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$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$ which in our case …

I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I". Statement: …

I think the following should work: Let $M$ be a compact manifold (just to be safe) and $\pi :E \to M$ a vector bundle. Since $E$ carries an action of $\mathbb{R}^{\times}$ there's an invariant notion …

Let $M$ be a smooth manifold. I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on …

There's already a question about the same topic but I think its aim is different. Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely di …

There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the litera …

Let $M$ be a manifold and let $\nabla$ be a connection on its tangent bundle and let $(E, \tilde \nabla )$ be a vector bundle with connection. These two connections together extended uniquely to conn …

I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the co …

Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions. Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ ar …

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ge …

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely …

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{ …

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization: Let $\{X_j\} \in Vect(M)$ be a f …

Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$: $$0 \to \mathfrak{h} \to \mathfrak{g} \to \ma …

This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try. At the risk of repeating well known stuff I tried t …