191 reputation
6
bio website
location Silesia, Czech Republic
age 35
visits member for 2 years
seen Apr 1 at 7:02

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