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Graduate student in mathematics.


May
4
awarded  lie-groups
May
3
awarded  Nice Answer
Apr
28
comment Three linearly independent vector fields
There is not an isomorphism-- I have just proved that there isn't.
Apr
26
comment show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$
I meant the long exact sequence associated to a pair (see e.g. the l.e.s. at en.wikipedia.org/wiki/Cohomology_with_compact_support with $X, U$ and $Z$). Because you should be able to use that with $X = S^n, Z = pt$ and $U = X - Z \simeq \mathbb R^n$ to get your desired result.
Apr
26
comment show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$
Does Lee talk about the long exact sequence in compactly supported cohomology associated to an open (or closed) subset?
Apr
26
comment show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$
You can't use the fact that $\mathbb R^n$ is contractible since compactly supported cohomology is only a homotopy invariant for homotopies that are proper maps.
Apr
26
comment The kernel of a differential one-form
@Ilcapitano yes, that is correct (assuming of course $\theta$ vanishes only at $m$).
Apr
21
comment Three linearly independent vector fields
$S^1\times S^2$ is not isomorphic to $SO(3)$. One way to see this is the fundamental group of the former is $\mathbb Z$ while the fundamental group of $SO(3)$ is $\mathbb Z_2$. Maybe you thought this because there is a fibration $S^1 \to SO(3) \to S^2$. Nonetheless, $S^1\times S^2$ is parallelizable since all orientable 3-manifolds are.
Apr
19
comment The magnetic monopole and the Hopf bundle
If you think of $\mathbb R^4 - \{0\}$ as $\mathbb C^2 - \{0\}$ then it seems likely (I haven't done any computations) that $\omega$ is a connection form on the principal $\mathbb C^\times$ bundle $\mathbb C^2 - \{0\} \to \mathbb C P^1 \simeq S^2$.
Apr
16
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 $g = g'$?