When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?











up vote
42
down vote

favorite
9












Let $1 leq k < n$ be natural numbers. Given orthonormal vectors $u_1,dots,u_k$ in ${bf R}^n$, one can always find an additional unit vector $v in {bf R}^n$ that is orthogonal to the preceding $k$. My question is: under what conditions on $k,n$ is it possible to make $v$ depend continuously on $u_1,dots,u_k$, as the tuple $(u_1,dots,u_k)$ ranges over all possible orthonormal systems? (For my application I actually want smooth dependence, but I think that a continuous map can be averaged out to be smooth without difficulty.)



When $k=n-1$ then one can just pick the unique unit normal to the span of the $u_1,dots,u_k$ that is consistent with a chosen orientation on ${bf R}^n$ (i.e., take wedge product and then Hodge dual, or just cross product in the $(k,n)=(2,3)$ case). But I don't know what is going on in lower dimension. Intuitively it seems to me that if $n$ is much larger than $k$ then the problem is so underdetermined that there should be no topological obstructions (such as that provided by the Borsuk-Ulam theorem), but I don't have the experience in algebraic topology to make this intuition precise.



It would suffice to exhibit a global section of the normal bundle of the (oriented) Grassmannian $Gr(k,n)$, though I don't know how to calculate the space of such sections.










share|cite|improve this question






















  • It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
    – Loop Space
    Nov 5 at 17:41






  • 8




    Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
    – Loop Space
    Nov 6 at 7:21















up vote
42
down vote

favorite
9












Let $1 leq k < n$ be natural numbers. Given orthonormal vectors $u_1,dots,u_k$ in ${bf R}^n$, one can always find an additional unit vector $v in {bf R}^n$ that is orthogonal to the preceding $k$. My question is: under what conditions on $k,n$ is it possible to make $v$ depend continuously on $u_1,dots,u_k$, as the tuple $(u_1,dots,u_k)$ ranges over all possible orthonormal systems? (For my application I actually want smooth dependence, but I think that a continuous map can be averaged out to be smooth without difficulty.)



When $k=n-1$ then one can just pick the unique unit normal to the span of the $u_1,dots,u_k$ that is consistent with a chosen orientation on ${bf R}^n$ (i.e., take wedge product and then Hodge dual, or just cross product in the $(k,n)=(2,3)$ case). But I don't know what is going on in lower dimension. Intuitively it seems to me that if $n$ is much larger than $k$ then the problem is so underdetermined that there should be no topological obstructions (such as that provided by the Borsuk-Ulam theorem), but I don't have the experience in algebraic topology to make this intuition precise.



It would suffice to exhibit a global section of the normal bundle of the (oriented) Grassmannian $Gr(k,n)$, though I don't know how to calculate the space of such sections.










share|cite|improve this question






















  • It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
    – Loop Space
    Nov 5 at 17:41






  • 8




    Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
    – Loop Space
    Nov 6 at 7:21













up vote
42
down vote

favorite
9









up vote
42
down vote

favorite
9






9





Let $1 leq k < n$ be natural numbers. Given orthonormal vectors $u_1,dots,u_k$ in ${bf R}^n$, one can always find an additional unit vector $v in {bf R}^n$ that is orthogonal to the preceding $k$. My question is: under what conditions on $k,n$ is it possible to make $v$ depend continuously on $u_1,dots,u_k$, as the tuple $(u_1,dots,u_k)$ ranges over all possible orthonormal systems? (For my application I actually want smooth dependence, but I think that a continuous map can be averaged out to be smooth without difficulty.)



When $k=n-1$ then one can just pick the unique unit normal to the span of the $u_1,dots,u_k$ that is consistent with a chosen orientation on ${bf R}^n$ (i.e., take wedge product and then Hodge dual, or just cross product in the $(k,n)=(2,3)$ case). But I don't know what is going on in lower dimension. Intuitively it seems to me that if $n$ is much larger than $k$ then the problem is so underdetermined that there should be no topological obstructions (such as that provided by the Borsuk-Ulam theorem), but I don't have the experience in algebraic topology to make this intuition precise.



It would suffice to exhibit a global section of the normal bundle of the (oriented) Grassmannian $Gr(k,n)$, though I don't know how to calculate the space of such sections.










share|cite|improve this question













Let $1 leq k < n$ be natural numbers. Given orthonormal vectors $u_1,dots,u_k$ in ${bf R}^n$, one can always find an additional unit vector $v in {bf R}^n$ that is orthogonal to the preceding $k$. My question is: under what conditions on $k,n$ is it possible to make $v$ depend continuously on $u_1,dots,u_k$, as the tuple $(u_1,dots,u_k)$ ranges over all possible orthonormal systems? (For my application I actually want smooth dependence, but I think that a continuous map can be averaged out to be smooth without difficulty.)



When $k=n-1$ then one can just pick the unique unit normal to the span of the $u_1,dots,u_k$ that is consistent with a chosen orientation on ${bf R}^n$ (i.e., take wedge product and then Hodge dual, or just cross product in the $(k,n)=(2,3)$ case). But I don't know what is going on in lower dimension. Intuitively it seems to me that if $n$ is much larger than $k$ then the problem is so underdetermined that there should be no topological obstructions (such as that provided by the Borsuk-Ulam theorem), but I don't have the experience in algebraic topology to make this intuition precise.



It would suffice to exhibit a global section of the normal bundle of the (oriented) Grassmannian $Gr(k,n)$, though I don't know how to calculate the space of such sections.







at.algebraic-topology grassmannians orthogonal-matrices






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 5 at 15:40









Terry Tao

56.9k17246337




56.9k17246337












  • It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
    – Loop Space
    Nov 5 at 17:41






  • 8




    Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
    – Loop Space
    Nov 6 at 7:21


















  • It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
    – Loop Space
    Nov 5 at 17:41






  • 8




    Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
    – Loop Space
    Nov 6 at 7:21
















It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
– Loop Space
Nov 5 at 17:41




It's been a loooong time since I've thought about such things, but I think that the obstruction you want is the Euler class, on the Wikipedia page it says "Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section". So you'd need to look at the pull-back of the Euler class under the map $Gr_{n,k} to Gr_{n,n-k}$ which maps $E$ to its orthogonal complement. The issue is a little more complicated if you're starting with a basis as then you need to pull back further to the frame bundle.
– Loop Space
Nov 5 at 17:41




8




8




Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
– Loop Space
Nov 6 at 7:21




Just a comment on your intuition (now that you have a full answer). Having lots of choice doesn't make it any easier to find a continuous choice because the act of making a choice is not continuous. So after making lots of local choices, you are confronted with the problem of patching them together and that is where the obstructions lie.
– Loop Space
Nov 6 at 7:21










4 Answers
4






active

oldest

votes

















up vote
50
down vote



accepted










$defRR{mathbb{R}}$ This problem was solved by



Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.



Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$.



All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$.



The $(3,8)$ product was computed by



Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401



to be given by the formula
$$X(a,b,c) = -a (overline{b} c) + a (b cdot c) - b (c cdot a) + c (a cdot b)$$
where $cdot$ is dot product while multiplication with no symbol and $overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.






share|cite|improve this answer






























    up vote
    35
    down vote













    Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.






    share|cite|improve this answer



















    • 8




      The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
      – R. van Dobben de Bruyn
      Nov 5 at 15:58








    • 15




      Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
      – Terry Tao
      Nov 5 at 16:19






    • 9




      Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
      – Thomas Rot
      Nov 5 at 18:03








    • 1




      This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
      – Michael
      Nov 6 at 22:25








    • 3




      @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
      – Aloizio Macedo
      Nov 7 at 19:09




















    up vote
    21
    down vote













    The space of orthonormal $k$-frames in $mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $operatorname{Gr}^+(k, n) = SO(n)/(SO(k)times SO(n-k))$. Let $gamma_k to operatorname{Gr}^+(k, n)$ denote the tautological bundle and let $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ denote its orthogonal complement. As you indicated, a nowhere-zero section of $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ would give rise to a map that you desire. In fact, such a map arises this way if and only if it is $SO(k)$-invariant.



    The inner product on $mathbb{R}^n$ allows us to define the map $P mapsto P^{perp}$ which induces a diffeomorphism $f : operatorname{Gr}^+(k, n) to operatorname{Gr}^+(n-k, n)$. Under this diffeomorphism we have $f^*gamma_{n-k} cong gamma_k^{perp}$, so $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ admits a nowhere-zero section if and only if $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ does. Therefore, we would like to know the answer to the following question:




    For which values of $k$ and $n$ does $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ admit a nowhere-zero section?




    One necessary condition is that $w_{n-k}(gamma_{n-k}) = 0$. Said another way, if $w_{n-k}(gamma_{n-k}) neq 0$, then there is no $SO(k)$-invariant map.





    In a previous version of this answer, I stated what I thought was the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ - I was incorrect. From this mistake, it followed that for $1 < k < n - 1$, $w_{n-k}(gamma_{n-k}) neq 0$ and hence there were no $SO(k)$-invariant maps for these values of $k$. This conclusion is false; there is a counterexample when $k = 2$ and $n = 7$ as David E Speyer pointed out in the comments below.



    Somewhat surprisingly, the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ is not known in general, see this question. The values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ also seems to be unknown.





    When $k = n - 1$, you described such a map which is in fact $SO(n-1)$-invariant. By the above correspondence, such maps exist because $gamma_1 to operatorname{Gr}^+(1, n) = S^{n-1}$ is trivial as it is an orientable line bundle (alternatively, $gamma_1$ is trivialised by the Euler vector field).



    When $k = 1$, first note that $gamma_{n-1} to operatorname{Gr}^+(n - 1, n) = S^{n-1}$ is isomorphic to the tangent bundle of $S^{n-1}$:



    begin{align*}
    TS^{n-1} &cong Toperatorname{Gr}^+(n-1, n)\
    &cong operatorname{Hom}(gamma_{n-1}, gamma_{n-1}^{perp})\
    &cong gamma_{n-1}^*otimesgamma_{n-1}^{perp}\
    &cong gamma_{n-1}otimes f^*gamma_1\
    &cong gamma_{n-1}
    end{align*}



    where the last isomorphism uses the fact that $gamma_1$, and hence $f^*gamma_1$, is trivial. By Poincaré-Hopf, $TS^{n-1}$ admits a section if and only if $n$ is even. In this case, the map can be written down explicitly: $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (-v_2, v_1, dots, -v_n, v_{n-1})$. Identifying $mathbb{R}^n$ and $mathbb{C}^{n/2}$ via $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (v_1 + iv_2, dots, v_{n-1} + iv_n)$, the aforementioned map is nothing but multiplication by $i$.



    Note, requiring $SO(k)$-invariance for $k = 1$ is not a restriction as $SO(1)$ is the trivial group.






    share|cite|improve this answer























    • It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
      – David E Speyer
      Nov 6 at 16:02








    • 1




      This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
      – David E Speyer
      Nov 6 at 16:02










    • @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
      – Michael Albanese
      Nov 6 at 16:32






    • 2




      @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
      – Michael Albanese
      Nov 6 at 17:37










    • @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
      – Michael Albanese
      Nov 8 at 23:10




















    up vote
    17
    down vote













    Denoting the Stiefel manifold of orthonormal $k$-frames in $mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
    $$
    S^{n-k-1}to V(k+1,n)to V(k,n),
    $$

    where the projection takes a $(k+1)$-frame to its first $k$ vectors.



    According to the paper



    Čadek, Martin; Mimura, Mamoru; Vanžura, Jiří, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43, No. 2, 411-428 (2003). ZBL1061.55015,



    the Euler class of this bundle is zero when $n-k$ is odd, and nonzero when $n-k$ is even.



    This extends Michael Albanese's answer slightly, showing that the maps cannot exist when $n-k$ is even with $1<k<n$, even without requiring $SO(k)$-invariance. I don't know if the bundle admits a section when $n-k$ is odd. The primary obstruction vanishes in these cases, but note that there may be higher obstructions in the groups $H^{i+1}(V(k,n);pi_i(S^{n-k-1}))$.






    share|cite|improve this answer























      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "504"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














       

      draft saved


      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f314613%2fwhen-can-one-continuously-prescribe-a-unit-vector-orthogonal-to-a-given-orthonor%23new-answer', 'question_page');
      }
      );

      Post as a guest
































      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      50
      down vote



      accepted










      $defRR{mathbb{R}}$ This problem was solved by



      Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.



      Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$.



      All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$.



      The $(3,8)$ product was computed by



      Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401



      to be given by the formula
      $$X(a,b,c) = -a (overline{b} c) + a (b cdot c) - b (c cdot a) + c (a cdot b)$$
      where $cdot$ is dot product while multiplication with no symbol and $overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.






      share|cite|improve this answer



























        up vote
        50
        down vote



        accepted










        $defRR{mathbb{R}}$ This problem was solved by



        Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.



        Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$.



        All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$.



        The $(3,8)$ product was computed by



        Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401



        to be given by the formula
        $$X(a,b,c) = -a (overline{b} c) + a (b cdot c) - b (c cdot a) + c (a cdot b)$$
        where $cdot$ is dot product while multiplication with no symbol and $overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.






        share|cite|improve this answer

























          up vote
          50
          down vote



          accepted







          up vote
          50
          down vote



          accepted






          $defRR{mathbb{R}}$ This problem was solved by



          Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.



          Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$.



          All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$.



          The $(3,8)$ product was computed by



          Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401



          to be given by the formula
          $$X(a,b,c) = -a (overline{b} c) + a (b cdot c) - b (c cdot a) + c (a cdot b)$$
          where $cdot$ is dot product while multiplication with no symbol and $overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.






          share|cite|improve this answer














          $defRR{mathbb{R}}$ This problem was solved by



          Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.



          Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$.



          All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$.



          The $(3,8)$ product was computed by



          Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401



          to be given by the formula
          $$X(a,b,c) = -a (overline{b} c) + a (b cdot c) - b (c cdot a) + c (a cdot b)$$
          where $cdot$ is dot product while multiplication with no symbol and $overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 6 at 20:27

























          answered Nov 6 at 14:39









          David E Speyer

          104k8269530




          104k8269530






















              up vote
              35
              down vote













              Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.






              share|cite|improve this answer



















              • 8




                The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
                – R. van Dobben de Bruyn
                Nov 5 at 15:58








              • 15




                Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
                – Terry Tao
                Nov 5 at 16:19






              • 9




                Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
                – Thomas Rot
                Nov 5 at 18:03








              • 1




                This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
                – Michael
                Nov 6 at 22:25








              • 3




                @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
                – Aloizio Macedo
                Nov 7 at 19:09

















              up vote
              35
              down vote













              Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.






              share|cite|improve this answer



















              • 8




                The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
                – R. van Dobben de Bruyn
                Nov 5 at 15:58








              • 15




                Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
                – Terry Tao
                Nov 5 at 16:19






              • 9




                Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
                – Thomas Rot
                Nov 5 at 18:03








              • 1




                This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
                – Michael
                Nov 6 at 22:25








              • 3




                @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
                – Aloizio Macedo
                Nov 7 at 19:09















              up vote
              35
              down vote










              up vote
              35
              down vote









              Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.






              share|cite|improve this answer














              Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Nov 5 at 17:54









              Alexandre Eremenko

              47.8k6131245




              47.8k6131245










              answered Nov 5 at 15:48









              Will Brian

              8,41923753




              8,41923753








              • 8




                The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
                – R. van Dobben de Bruyn
                Nov 5 at 15:58








              • 15




                Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
                – Terry Tao
                Nov 5 at 16:19






              • 9




                Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
                – Thomas Rot
                Nov 5 at 18:03








              • 1




                This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
                – Michael
                Nov 6 at 22:25








              • 3




                @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
                – Aloizio Macedo
                Nov 7 at 19:09
















              • 8




                The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
                – R. van Dobben de Bruyn
                Nov 5 at 15:58








              • 15




                Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
                – Terry Tao
                Nov 5 at 16:19






              • 9




                Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
                – Thomas Rot
                Nov 5 at 18:03








              • 1




                This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
                – Michael
                Nov 6 at 22:25








              • 3




                @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
                – Aloizio Macedo
                Nov 7 at 19:09










              8




              8




              The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
              – R. van Dobben de Bruyn
              Nov 5 at 15:58






              The hairy ball theorem actually applies to any even-dimensional sphere, so this would also settle the case $n > 1$ odd and $k = 1$.
              – R. van Dobben de Bruyn
              Nov 5 at 15:58






              15




              15




              Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
              – Terry Tao
              Nov 5 at 16:19




              Wow, I can't believe I had forgotten about this theorem. So my intuition that the problem is too underdetermined to have an obstruction in high dimension is wrong, apparently.
              – Terry Tao
              Nov 5 at 16:19




              9




              9




              Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
              – Thomas Rot
              Nov 5 at 18:03






              Write $n=2^{c+4d} a$, with $a$ odd and $0leq c leq 3$. Then there are $2^c+8 d -1$ linear independent sections of the tangent bundle of the sphere, which you can make orthogonal I think. This is a famous theorem of Adams.
              – Thomas Rot
              Nov 5 at 18:03






              1




              1




              This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
              – Michael
              Nov 6 at 22:25






              This is a terrific answer for $mathbb{R}^3-{0}$, but I'm not sure that it applies to $mathbb{R}^3$, for there's no non-degenerate vector field in $mathbb{R}^3$ that would be normal to the sphere.
              – Michael
              Nov 6 at 22:25






              3




              3




              @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
              – Aloizio Macedo
              Nov 7 at 19:09






              @Michael The question is not asking for complementation of a vector field in $mathbb{R}^n$. It is asking for complementation when "the tuple ranges over all possible orthonormal systems". This is precisely the sphere $S^2$ when $n=3, k=1$.
              – Aloizio Macedo
              Nov 7 at 19:09












              up vote
              21
              down vote













              The space of orthonormal $k$-frames in $mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $operatorname{Gr}^+(k, n) = SO(n)/(SO(k)times SO(n-k))$. Let $gamma_k to operatorname{Gr}^+(k, n)$ denote the tautological bundle and let $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ denote its orthogonal complement. As you indicated, a nowhere-zero section of $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ would give rise to a map that you desire. In fact, such a map arises this way if and only if it is $SO(k)$-invariant.



              The inner product on $mathbb{R}^n$ allows us to define the map $P mapsto P^{perp}$ which induces a diffeomorphism $f : operatorname{Gr}^+(k, n) to operatorname{Gr}^+(n-k, n)$. Under this diffeomorphism we have $f^*gamma_{n-k} cong gamma_k^{perp}$, so $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ admits a nowhere-zero section if and only if $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ does. Therefore, we would like to know the answer to the following question:




              For which values of $k$ and $n$ does $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ admit a nowhere-zero section?




              One necessary condition is that $w_{n-k}(gamma_{n-k}) = 0$. Said another way, if $w_{n-k}(gamma_{n-k}) neq 0$, then there is no $SO(k)$-invariant map.





              In a previous version of this answer, I stated what I thought was the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ - I was incorrect. From this mistake, it followed that for $1 < k < n - 1$, $w_{n-k}(gamma_{n-k}) neq 0$ and hence there were no $SO(k)$-invariant maps for these values of $k$. This conclusion is false; there is a counterexample when $k = 2$ and $n = 7$ as David E Speyer pointed out in the comments below.



              Somewhat surprisingly, the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ is not known in general, see this question. The values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ also seems to be unknown.





              When $k = n - 1$, you described such a map which is in fact $SO(n-1)$-invariant. By the above correspondence, such maps exist because $gamma_1 to operatorname{Gr}^+(1, n) = S^{n-1}$ is trivial as it is an orientable line bundle (alternatively, $gamma_1$ is trivialised by the Euler vector field).



              When $k = 1$, first note that $gamma_{n-1} to operatorname{Gr}^+(n - 1, n) = S^{n-1}$ is isomorphic to the tangent bundle of $S^{n-1}$:



              begin{align*}
              TS^{n-1} &cong Toperatorname{Gr}^+(n-1, n)\
              &cong operatorname{Hom}(gamma_{n-1}, gamma_{n-1}^{perp})\
              &cong gamma_{n-1}^*otimesgamma_{n-1}^{perp}\
              &cong gamma_{n-1}otimes f^*gamma_1\
              &cong gamma_{n-1}
              end{align*}



              where the last isomorphism uses the fact that $gamma_1$, and hence $f^*gamma_1$, is trivial. By Poincaré-Hopf, $TS^{n-1}$ admits a section if and only if $n$ is even. In this case, the map can be written down explicitly: $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (-v_2, v_1, dots, -v_n, v_{n-1})$. Identifying $mathbb{R}^n$ and $mathbb{C}^{n/2}$ via $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (v_1 + iv_2, dots, v_{n-1} + iv_n)$, the aforementioned map is nothing but multiplication by $i$.



              Note, requiring $SO(k)$-invariance for $k = 1$ is not a restriction as $SO(1)$ is the trivial group.






              share|cite|improve this answer























              • It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
                – David E Speyer
                Nov 6 at 16:02








              • 1




                This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
                – David E Speyer
                Nov 6 at 16:02










              • @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
                – Michael Albanese
                Nov 6 at 16:32






              • 2




                @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
                – Michael Albanese
                Nov 6 at 17:37










              • @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
                – Michael Albanese
                Nov 8 at 23:10

















              up vote
              21
              down vote













              The space of orthonormal $k$-frames in $mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $operatorname{Gr}^+(k, n) = SO(n)/(SO(k)times SO(n-k))$. Let $gamma_k to operatorname{Gr}^+(k, n)$ denote the tautological bundle and let $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ denote its orthogonal complement. As you indicated, a nowhere-zero section of $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ would give rise to a map that you desire. In fact, such a map arises this way if and only if it is $SO(k)$-invariant.



              The inner product on $mathbb{R}^n$ allows us to define the map $P mapsto P^{perp}$ which induces a diffeomorphism $f : operatorname{Gr}^+(k, n) to operatorname{Gr}^+(n-k, n)$. Under this diffeomorphism we have $f^*gamma_{n-k} cong gamma_k^{perp}$, so $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ admits a nowhere-zero section if and only if $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ does. Therefore, we would like to know the answer to the following question:




              For which values of $k$ and $n$ does $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ admit a nowhere-zero section?




              One necessary condition is that $w_{n-k}(gamma_{n-k}) = 0$. Said another way, if $w_{n-k}(gamma_{n-k}) neq 0$, then there is no $SO(k)$-invariant map.





              In a previous version of this answer, I stated what I thought was the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ - I was incorrect. From this mistake, it followed that for $1 < k < n - 1$, $w_{n-k}(gamma_{n-k}) neq 0$ and hence there were no $SO(k)$-invariant maps for these values of $k$. This conclusion is false; there is a counterexample when $k = 2$ and $n = 7$ as David E Speyer pointed out in the comments below.



              Somewhat surprisingly, the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ is not known in general, see this question. The values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ also seems to be unknown.





              When $k = n - 1$, you described such a map which is in fact $SO(n-1)$-invariant. By the above correspondence, such maps exist because $gamma_1 to operatorname{Gr}^+(1, n) = S^{n-1}$ is trivial as it is an orientable line bundle (alternatively, $gamma_1$ is trivialised by the Euler vector field).



              When $k = 1$, first note that $gamma_{n-1} to operatorname{Gr}^+(n - 1, n) = S^{n-1}$ is isomorphic to the tangent bundle of $S^{n-1}$:



              begin{align*}
              TS^{n-1} &cong Toperatorname{Gr}^+(n-1, n)\
              &cong operatorname{Hom}(gamma_{n-1}, gamma_{n-1}^{perp})\
              &cong gamma_{n-1}^*otimesgamma_{n-1}^{perp}\
              &cong gamma_{n-1}otimes f^*gamma_1\
              &cong gamma_{n-1}
              end{align*}



              where the last isomorphism uses the fact that $gamma_1$, and hence $f^*gamma_1$, is trivial. By Poincaré-Hopf, $TS^{n-1}$ admits a section if and only if $n$ is even. In this case, the map can be written down explicitly: $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (-v_2, v_1, dots, -v_n, v_{n-1})$. Identifying $mathbb{R}^n$ and $mathbb{C}^{n/2}$ via $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (v_1 + iv_2, dots, v_{n-1} + iv_n)$, the aforementioned map is nothing but multiplication by $i$.



              Note, requiring $SO(k)$-invariance for $k = 1$ is not a restriction as $SO(1)$ is the trivial group.






              share|cite|improve this answer























              • It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
                – David E Speyer
                Nov 6 at 16:02








              • 1




                This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
                – David E Speyer
                Nov 6 at 16:02










              • @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
                – Michael Albanese
                Nov 6 at 16:32






              • 2




                @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
                – Michael Albanese
                Nov 6 at 17:37










              • @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
                – Michael Albanese
                Nov 8 at 23:10















              up vote
              21
              down vote










              up vote
              21
              down vote









              The space of orthonormal $k$-frames in $mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $operatorname{Gr}^+(k, n) = SO(n)/(SO(k)times SO(n-k))$. Let $gamma_k to operatorname{Gr}^+(k, n)$ denote the tautological bundle and let $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ denote its orthogonal complement. As you indicated, a nowhere-zero section of $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ would give rise to a map that you desire. In fact, such a map arises this way if and only if it is $SO(k)$-invariant.



              The inner product on $mathbb{R}^n$ allows us to define the map $P mapsto P^{perp}$ which induces a diffeomorphism $f : operatorname{Gr}^+(k, n) to operatorname{Gr}^+(n-k, n)$. Under this diffeomorphism we have $f^*gamma_{n-k} cong gamma_k^{perp}$, so $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ admits a nowhere-zero section if and only if $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ does. Therefore, we would like to know the answer to the following question:




              For which values of $k$ and $n$ does $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ admit a nowhere-zero section?




              One necessary condition is that $w_{n-k}(gamma_{n-k}) = 0$. Said another way, if $w_{n-k}(gamma_{n-k}) neq 0$, then there is no $SO(k)$-invariant map.





              In a previous version of this answer, I stated what I thought was the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ - I was incorrect. From this mistake, it followed that for $1 < k < n - 1$, $w_{n-k}(gamma_{n-k}) neq 0$ and hence there were no $SO(k)$-invariant maps for these values of $k$. This conclusion is false; there is a counterexample when $k = 2$ and $n = 7$ as David E Speyer pointed out in the comments below.



              Somewhat surprisingly, the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ is not known in general, see this question. The values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ also seems to be unknown.





              When $k = n - 1$, you described such a map which is in fact $SO(n-1)$-invariant. By the above correspondence, such maps exist because $gamma_1 to operatorname{Gr}^+(1, n) = S^{n-1}$ is trivial as it is an orientable line bundle (alternatively, $gamma_1$ is trivialised by the Euler vector field).



              When $k = 1$, first note that $gamma_{n-1} to operatorname{Gr}^+(n - 1, n) = S^{n-1}$ is isomorphic to the tangent bundle of $S^{n-1}$:



              begin{align*}
              TS^{n-1} &cong Toperatorname{Gr}^+(n-1, n)\
              &cong operatorname{Hom}(gamma_{n-1}, gamma_{n-1}^{perp})\
              &cong gamma_{n-1}^*otimesgamma_{n-1}^{perp}\
              &cong gamma_{n-1}otimes f^*gamma_1\
              &cong gamma_{n-1}
              end{align*}



              where the last isomorphism uses the fact that $gamma_1$, and hence $f^*gamma_1$, is trivial. By Poincaré-Hopf, $TS^{n-1}$ admits a section if and only if $n$ is even. In this case, the map can be written down explicitly: $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (-v_2, v_1, dots, -v_n, v_{n-1})$. Identifying $mathbb{R}^n$ and $mathbb{C}^{n/2}$ via $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (v_1 + iv_2, dots, v_{n-1} + iv_n)$, the aforementioned map is nothing but multiplication by $i$.



              Note, requiring $SO(k)$-invariance for $k = 1$ is not a restriction as $SO(1)$ is the trivial group.






              share|cite|improve this answer














              The space of orthonormal $k$-frames in $mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $operatorname{Gr}^+(k, n) = SO(n)/(SO(k)times SO(n-k))$. Let $gamma_k to operatorname{Gr}^+(k, n)$ denote the tautological bundle and let $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ denote its orthogonal complement. As you indicated, a nowhere-zero section of $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ would give rise to a map that you desire. In fact, such a map arises this way if and only if it is $SO(k)$-invariant.



              The inner product on $mathbb{R}^n$ allows us to define the map $P mapsto P^{perp}$ which induces a diffeomorphism $f : operatorname{Gr}^+(k, n) to operatorname{Gr}^+(n-k, n)$. Under this diffeomorphism we have $f^*gamma_{n-k} cong gamma_k^{perp}$, so $gamma_k^{perp} to operatorname{Gr}^+(k, n)$ admits a nowhere-zero section if and only if $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ does. Therefore, we would like to know the answer to the following question:




              For which values of $k$ and $n$ does $gamma_{n-k} to operatorname{Gr}^+(n-k, n)$ admit a nowhere-zero section?




              One necessary condition is that $w_{n-k}(gamma_{n-k}) = 0$. Said another way, if $w_{n-k}(gamma_{n-k}) neq 0$, then there is no $SO(k)$-invariant map.





              In a previous version of this answer, I stated what I thought was the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ - I was incorrect. From this mistake, it followed that for $1 < k < n - 1$, $w_{n-k}(gamma_{n-k}) neq 0$ and hence there were no $SO(k)$-invariant maps for these values of $k$. This conclusion is false; there is a counterexample when $k = 2$ and $n = 7$ as David E Speyer pointed out in the comments below.



              Somewhat surprisingly, the $mathbb{Z}_2$ cohomology ring of $operatorname{Gr}^+(k, n)$ is not known in general, see this question. The values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ also seems to be unknown.





              When $k = n - 1$, you described such a map which is in fact $SO(n-1)$-invariant. By the above correspondence, such maps exist because $gamma_1 to operatorname{Gr}^+(1, n) = S^{n-1}$ is trivial as it is an orientable line bundle (alternatively, $gamma_1$ is trivialised by the Euler vector field).



              When $k = 1$, first note that $gamma_{n-1} to operatorname{Gr}^+(n - 1, n) = S^{n-1}$ is isomorphic to the tangent bundle of $S^{n-1}$:



              begin{align*}
              TS^{n-1} &cong Toperatorname{Gr}^+(n-1, n)\
              &cong operatorname{Hom}(gamma_{n-1}, gamma_{n-1}^{perp})\
              &cong gamma_{n-1}^*otimesgamma_{n-1}^{perp}\
              &cong gamma_{n-1}otimes f^*gamma_1\
              &cong gamma_{n-1}
              end{align*}



              where the last isomorphism uses the fact that $gamma_1$, and hence $f^*gamma_1$, is trivial. By Poincaré-Hopf, $TS^{n-1}$ admits a section if and only if $n$ is even. In this case, the map can be written down explicitly: $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (-v_2, v_1, dots, -v_n, v_{n-1})$. Identifying $mathbb{R}^n$ and $mathbb{C}^{n/2}$ via $(v_1, v_2, dots, v_{n-1}, v_n) mapsto (v_1 + iv_2, dots, v_{n-1} + iv_n)$, the aforementioned map is nothing but multiplication by $i$.



              Note, requiring $SO(k)$-invariance for $k = 1$ is not a restriction as $SO(1)$ is the trivial group.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Nov 8 at 23:09

























              answered Nov 6 at 3:16









              Michael Albanese

              7,29054889




              7,29054889












              • It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
                – David E Speyer
                Nov 6 at 16:02








              • 1




                This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
                – David E Speyer
                Nov 6 at 16:02










              • @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
                – Michael Albanese
                Nov 6 at 16:32






              • 2




                @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
                – Michael Albanese
                Nov 6 at 17:37










              • @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
                – Michael Albanese
                Nov 8 at 23:10




















              • It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
                – David E Speyer
                Nov 6 at 16:02








              • 1




                This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
                – David E Speyer
                Nov 6 at 16:02










              • @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
                – Michael Albanese
                Nov 6 at 16:32






              • 2




                @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
                – Michael Albanese
                Nov 6 at 17:37










              • @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
                – Michael Albanese
                Nov 8 at 23:10


















              It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
              – David E Speyer
              Nov 6 at 16:02






              It seems to me that something is wrong with this argument, because of the existence of the 7 dimensional cross product: en.wikipedia.org/wiki/Seven-dimensional_cross_product . This is an antisymmetric bilinear map $R^7 times R^7 to R^7$ such that $u times v$ is always perpendicular to $u$ and $v$ and, if $u$ and $v$ are orthogonal, then $|u times v| = |u| |v|$. Using bilinearity and antisymmetry, I get $(cos theta u + sin theta v) times (-sin theta u + cos theta v) = (cos^2 theta + sin^2 theta) (u times v) = u times v$, meaning this map is $SO(2)$ invariant.
              – David E Speyer
              Nov 6 at 16:02






              1




              1




              This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
              – David E Speyer
              Nov 6 at 16:02




              This would seem to give a section of $gamma_2^{perp} to G^+(2,7)$. What did I miss?
              – David E Speyer
              Nov 6 at 16:02












              @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
              – Michael Albanese
              Nov 6 at 16:32




              @DavidESpeyer: I can't see anything wrong with what you've written, so there must be something wrong with my answer. Maybe the cohomology ring of $operatorname{Gr}^+(n-k, n)$ is not correct. I don't know of a reference, but I was under the impression that this was correct.
              – Michael Albanese
              Nov 6 at 16:32




              2




              2




              @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
              – Michael Albanese
              Nov 6 at 17:37




              @DavidESpeyer: The cohomology ring is wrong as can be seen in the case $k = 2$, $n = 4$ where $operatorname{Gr}^+(2, 4) = S^2times S^2$. I will try to think about this and see what can be salvaged.
              – Michael Albanese
              Nov 6 at 17:37












              @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
              – Michael Albanese
              Nov 8 at 23:10






              @DavidESpeyer: It seems that the values of $k$ and $n$ for which $w_{n-k}(gamma_{n-k}) neq 0$ is not known; see this question.
              – Michael Albanese
              Nov 8 at 23:10












              up vote
              17
              down vote













              Denoting the Stiefel manifold of orthonormal $k$-frames in $mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
              $$
              S^{n-k-1}to V(k+1,n)to V(k,n),
              $$

              where the projection takes a $(k+1)$-frame to its first $k$ vectors.



              According to the paper



              Čadek, Martin; Mimura, Mamoru; Vanžura, Jiří, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43, No. 2, 411-428 (2003). ZBL1061.55015,



              the Euler class of this bundle is zero when $n-k$ is odd, and nonzero when $n-k$ is even.



              This extends Michael Albanese's answer slightly, showing that the maps cannot exist when $n-k$ is even with $1<k<n$, even without requiring $SO(k)$-invariance. I don't know if the bundle admits a section when $n-k$ is odd. The primary obstruction vanishes in these cases, but note that there may be higher obstructions in the groups $H^{i+1}(V(k,n);pi_i(S^{n-k-1}))$.






              share|cite|improve this answer



























                up vote
                17
                down vote













                Denoting the Stiefel manifold of orthonormal $k$-frames in $mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
                $$
                S^{n-k-1}to V(k+1,n)to V(k,n),
                $$

                where the projection takes a $(k+1)$-frame to its first $k$ vectors.



                According to the paper



                Čadek, Martin; Mimura, Mamoru; Vanžura, Jiří, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43, No. 2, 411-428 (2003). ZBL1061.55015,



                the Euler class of this bundle is zero when $n-k$ is odd, and nonzero when $n-k$ is even.



                This extends Michael Albanese's answer slightly, showing that the maps cannot exist when $n-k$ is even with $1<k<n$, even without requiring $SO(k)$-invariance. I don't know if the bundle admits a section when $n-k$ is odd. The primary obstruction vanishes in these cases, but note that there may be higher obstructions in the groups $H^{i+1}(V(k,n);pi_i(S^{n-k-1}))$.






                share|cite|improve this answer

























                  up vote
                  17
                  down vote










                  up vote
                  17
                  down vote









                  Denoting the Stiefel manifold of orthonormal $k$-frames in $mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
                  $$
                  S^{n-k-1}to V(k+1,n)to V(k,n),
                  $$

                  where the projection takes a $(k+1)$-frame to its first $k$ vectors.



                  According to the paper



                  Čadek, Martin; Mimura, Mamoru; Vanžura, Jiří, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43, No. 2, 411-428 (2003). ZBL1061.55015,



                  the Euler class of this bundle is zero when $n-k$ is odd, and nonzero when $n-k$ is even.



                  This extends Michael Albanese's answer slightly, showing that the maps cannot exist when $n-k$ is even with $1<k<n$, even without requiring $SO(k)$-invariance. I don't know if the bundle admits a section when $n-k$ is odd. The primary obstruction vanishes in these cases, but note that there may be higher obstructions in the groups $H^{i+1}(V(k,n);pi_i(S^{n-k-1}))$.






                  share|cite|improve this answer














                  Denoting the Stiefel manifold of orthonormal $k$-frames in $mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
                  $$
                  S^{n-k-1}to V(k+1,n)to V(k,n),
                  $$

                  where the projection takes a $(k+1)$-frame to its first $k$ vectors.



                  According to the paper



                  Čadek, Martin; Mimura, Mamoru; Vanžura, Jiří, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43, No. 2, 411-428 (2003). ZBL1061.55015,



                  the Euler class of this bundle is zero when $n-k$ is odd, and nonzero when $n-k$ is even.



                  This extends Michael Albanese's answer slightly, showing that the maps cannot exist when $n-k$ is even with $1<k<n$, even without requiring $SO(k)$-invariance. I don't know if the bundle admits a section when $n-k$ is odd. The primary obstruction vanishes in these cases, but note that there may be higher obstructions in the groups $H^{i+1}(V(k,n);pi_i(S^{n-k-1}))$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Nov 6 at 11:56

























                  answered Nov 6 at 7:52









                  Mark Grant

                  21.4k654128




                  21.4k654128






























                       

                      draft saved


                      draft discarded



















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f314613%2fwhen-can-one-continuously-prescribe-a-unit-vector-orthogonal-to-a-given-orthonor%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest




















































































                      Popular posts from this blog

                      Guess what letter conforming each word

                      Port of Spain

                      Run scheduled task as local user group (not BUILTIN)