Eric O. Korman
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 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 $g = g'$? 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 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)? Apr 9 comment Complex line bundle at symplectic manifold A line bundle and connection with curvature $\omega$ exist if and only if $[\omega/2\pi] \in H^2(M;\mathbb Z)$. Apr 9 comment Explicit form of the foliation associated to a differential one-form The leaves are the submanifolds of the form $\{(x^0_0 \exp(t), x^1_0 \exp(t), x^2, x^3) \mid t, x^2,x^3 \in \mathbb R\}$ for some constants $x^0_0, x^1_0$.