bio | website | |
---|---|---|
location | Silesia, Czech Republic | |
age | 35 | |
visits | member for | 2 years |
seen | Apr 18 at 5:15 | |
stats | profile views | 392 |
Reasonably expert in Differential Geometry, I work mostly with jet bundles and their natural structures trying to exploit them in the areas of nonlinear PDEs and Calculus of Variations.
Nov 25 |
comment |
Space of differential operators
Just noticed a minor bug in my answer: $\widetilde{\mathrm{Diff}}(A,B)$ should be defined as $C^\infty(M\times[0,1])\otimes_{C^\infty(M)}\mathrm{Diff}(A,B)$. The rest is unaltered. |
Nov 25 |
answered | Space of differential operators |
Nov 18 |
comment |
Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?
Dear GiuseppeTortorella, here's my own point of view on non-coorientability: if $N=\mathbb{P}T^*_xM$, then the line bundle $(T(\mathbb{P}T^*M))/C$, restricted to $N$, is nothing but the tautological line bundle $\ell$ over the projective space $\mathbb{P}^{n-1}$, which is non-trivial (by $n$ I mean the dimension of $M$). (If $\ell$ is trivial, then it admits a nowhere zero section which, in turn, allows to embed $\mathbb{P}^{n-1}$ into $\mathbb{R}^n$.) P.S. Thanks to @Petya for suggesting the counterexample. |
Sep 10 |
comment |
Legendrian Tubular Neighborhood Theorem
You just confirmed what I have guessed in my answer below: but we still do not know whether the contact structure on this neighborhood of $L$ is the restriction of the contact structure on the environment (called $(Y,\lambda)$ in the question). |
Sep 10 |
answered | Legendrian Tubular Neighborhood Theorem |
Sep 4 |
comment |
A hypersurface in the Grassmannian of endomorphisms.
Did you try to see how your polynomial looks like if you Plücker-embed your Grassmannian into $\mathbb{P}\Lambda^{2q}{\frak{sl}}(V)$? I guess you'd discover something related to the Lie algebra cohomology of ${\frak sl}(V)$ and/or its universal enveloping algebra. In fact, if we were in the real case, I would say that $E$ corresonds to a left-invariant distribution on the Lie group $SL(V)$, and that your polynomial captures some integrability properties of $E$. |
Sep 3 |
comment |
When a hyperplane of symmetric forms is determined by a quadric hypersurface?
@LevBorisov: at a first sight it looks like one needs a splitting to make it canonical (for any dimension of $L$), but I'm convinced that, in the end, the contributions of the splittings cancel out (one needs two of them). My idea is to use the short exact sequence $0\to L\to V\to L^*\to 0$ associated to any Lagrangian subspace (here $L^*=V/L$ is NOT a subspace of $V$), whose dual is $0\to L\to V^*\to L^*\to 0$, and try to combine $q$ and $q^*$ (acting crosswise between the endpoints) into a linear map between the central points of the sequences... but this is precisely the point I'm stuck to! |
Sep 3 |
asked | When a hyperplane of symmetric forms is determined by a quadric hypersurface? |
Sep 2 |
comment |
Fibred manifolds with boundary
Understood as a submanifold, the boundary $\partial M$ of $M$ has co-dimension 1: accordingly, the tangent bundle of the former should be a submanifold of co-dimension 2 of the latter. Hence, the category of manifolds with boundary is not rich enough for your purposes: you may give a look at Michor's "Manifolds of Differential Mappings", which provides a solid framework for this sort of objects. E.g., in ¶2.6 you will find the definition of the tangent bundle to a "manifold with corners". |
Aug 30 |
answered | Does Frobenius theorem apply to vector-valued function? |
Aug 28 |
comment |
Does Frobenius theorem apply to vector-valued function?
Your example is about the closedness of the differential 1-form $\omega=u_1dx_1+\ldots+u_ndx_n$, which has the solution $u_i=\frac{\partial F}{\partial x_i}$, with arbitrary $F$. Concerning the general question, I would think in terms of differential consequences and prolongations (see, e.g., Kuranishi's "On E. Cartan's prolongation theorem of exterior differential systems."): in this context, "integrability" means that the prolonged system projects back to the whole original one (you may check that this reduces to the Frobenius theorem in the case of a PDE given by vector fields). |
Aug 23 |
comment |
An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form
Reading your answer made me wonder: is there any natural projection $\Lambda^3(S^2 V^*)\to S^2(\Lambda^3 V^*)$? (By $\Lambda$ and $S$ I mean skew-symmetric and symmetric tensors, respectively.) If yes, then we are dealing with something like $f^2=c\sigma^3$: when $V$ is 3D, then (the square of) a nonzero $f$ spans the whole $S^2(\Lambda^3 V^*)$ and then the positive answer follows from the surjectivity of the above map. But when $V$ is larger, not all the $\sigma^3$ will project on $f^2$, and one must impose the condition you've explained. |
Aug 20 |
comment |
extending a vector bundle
Generalizing @DamianRössler's remark: if the embedding $N\subseteq M$ is a deformation retract, i.e., it admits a homotopy inverse, then the extension is possible. E.g., you can extend $TS^2$ to $\mathbb{R}^3\smallsetminus\{0\}$. |
Aug 7 |
comment |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
@robot: I did not know about "Penrose transform" but, after a quick look at the book you've suggested, I realized that it deals with the kind of problems mentioned by Francois Ziegler at the end of his comment; I'll certainly deepen the topic, though I'd still like to see an example concerning submanifolds! (now it is too late to apply for DGA: if by "expert" you mean Lychagin, then I've just spoken to him about my problem; if you mean "Landsberg", I'm going to email him right now... Did you have someboby elese in your mind?) |
Aug 6 |
revised |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
Added a reformulation of the main question, which takes into consideration the comments and partial answers received so far |
Aug 5 |
comment |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
... in which case these $\alpha$-surfaces are precisely the "double fibration transforms" (see Francois' answer) $\nu(\lambda^{-1}(\{H\}))$ of a singleton in $G_{k-1}(\mathbb{R}^n)$. I agree: tangency to the $\alpha$-distribution can be used to detect a double filtration transfom of a 0-dimensional manifold (I had figured out this by myself), but what about the next cases? |
Aug 5 |
comment |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
Is this "incidence cone" precisely the Segre variety $\mathbb{P}(\zeta^*)\times\mathbb{P}(\mathbb{R}^n/\zeta)$ in $\mathbb{P}(T_\zeta G_k(\mathbb{R}^n))$? If yes, then these $\alpha$-planes and $\beta$-planes should be the fibers of the cartesian product $\mathbb{P}(\zeta^*)\times\mathbb{P}(\mathbb{R}^n/\zeta)$. I have thought about this approach, but on the flag variety $F_{k-1,k}(\mathbb{R}^n)$, where the $\alpha$-planes correspond to an actual distribution. Are you sure that the $\alpha$-surfaces are made of $k$-planes containing a common line? I would say a common $k-1$-plane... |
Aug 5 |
comment |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
You provided me with valuable information indeed! I was aware of the "double fibration" you've depicted above from a question answered by Robert Bryant: mathoverflow.net/questions/96055/…. He also mentioned a canonical distribution on $F_{l,n}(V)$ obtained as the direct sum of the $\lambda$- and $\nu$-vertical ones, which I found very useful for my work, but he would not tell me who discovered it first! Now I'm in trouble because I need to use it and I don't even know how to call it! Do you have any clue? |
Aug 5 |
comment |
Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces
Thanks to @Francois Ziegler's partial answer now I have the keywords I was looking for, plus references! I'm going to go through the (huge) literature on this topic... but I'd be grateful if someone showed me even a simple example of characterization of the image of a double fibration transform (thus answering the Actual Question). |
Aug 4 |
asked | Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces |