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Graduate student in mathematics.


2d
comment Rigidity for Lie Groups
why does every auto of $\tilde G$ restrict to an auto of $\Gamma$? Did you mean $G$ instead of $\tilde G$ there?
Apr
14
comment Books in spectral theory for finite dimensional spaces
I'm not an expert but I don't think spectral theory for finite dimensional spaces extends to anything beyond the basics of eigenspaces and eigenvectors.
Apr
14
comment Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.
A very general theorem that tells you this is a locally trivial fibration is Ehresmann's theorem: en.wikipedia.org/wiki/Ehresmann's_theorem
Apr
12
comment Notation for the Covariant Derivative of a smooth section in the direction of a tangent vector X
I think you should answer your own question and then accept it.
Apr
12
answered The kernel of a differential one-form
Apr
12
revised How to differentiate this
deleted 181 characters in body; edited tags
Apr
11
comment What is the kernel of a Maurer-Cartan form?
Oh ok. It looks like they're using a much more general definition of Maurer-Cartan form (for example, their M.C. forms are not even on a Lie group necessarily). It seems like the prototypical example of such a form is a flat connection on a principal $G$-bundle. Note that the kernel of a connection defines a horizontal distribution and this distribution is integrable (and therefore gives rise to a foliation on the total space of the principal bundle) if and only if the connection is flat.
Apr
11
comment What is the kernel of a Maurer-Cartan form?
But do you have an example of where this appears in the foliation literature you mention?
Apr
11
comment What is the kernel of a Maurer-Cartan form?
For 1) It is a vector space valued 1-form so it gives a linear map from vector fields to the Lie algebra. Do you have more context for 2)? I don't see it can have a kernel (viewing it as a map as in 1)) since it takes a tangent vector $v$ at a point $g$ to ${L_{g^{-1}}}_* v$ which is non-zero if $v$ is non-zero since left multiplicaiton by $g$ is a diffeomorphism.
Apr
11
answered Isogenies and dimensions
Apr
10
comment Isogenies and dimensions
Or is your question if $g = g'$?
Apr
10
comment Isogenies and dimensions
Do you mean the linear map $\mathbb C^g \to \mathbb C^{g'}$ is surjective?
Apr
10
comment Can a ring isomorphism change the structure of a module?
Yes that should be true. Though the Clifford algebra is a simpler object then the group algebra of $SO(2n)$.
Apr
10
comment Can a ring isomorphism change the structure of a module?
Ok, I guess you can turn that into an example for a ring if you know about Clifford algebras: take the even Clifford algebra on $\mathbb R^{2n}$. Then the even and odd spinors are two non-isomorphic representations.
Apr
10
comment Can a ring isomorphism change the structure of a module?
Note that if $\phi$ is an inner automorphism then you get an isomorphic module, but in general you don't. I can't think of an immediate example for rings but for groups (where modules are representations) an example is that $Spin(2n)$ has two non-isomorphic spin representations but are related by the outer automorphism of $Spin(2n)$.
Apr
10
comment Notation in “Curvature in Mathematics and Physics” by Sternberg
$Y\times [0,h]$ is the set of all pairs $(y,t)$ with $y\in Y$ and $t \in [0,h]$.
Apr
10
revised Notation in “Curvature in Mathematics and Physics” by Sternberg
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Apr
10
comment Notation in “Curvature in Mathematics and Physics” by Sternberg
what part don't you understand? or maybe, what parts do you understand?
Apr
10
comment Notation for the Covariant Derivative of a smooth section in the direction of a tangent vector X
If $\alpha$ is a 1-form and $v$ a vector field, then does the notation $\langle \alpha, v\rangle$ make sense to you?
Apr
10
comment Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?
I'm guessing you want $M$ to be connected (or at least have finitely many connected components)?