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Actually, matrix-valued differential forms are used a lot in hypercomplex analysis/hypercomplex geometry, which, as the name suggests, includes certain complex manifolds. There is a nice account of su …

This is precisely Proposition 1.120 on p.49 in Besse's "Einstein Manifolds" (I am using the reprint of the 1987 edition, so the numbering may be different in the older edition): A 3-dimensional pseud …

More is true: if $X$ is a topological manifold, then in fact $\operatorname{Der}(C(X)) = 0$, where $C(X)$ denotes the $\mathbb{R}$-algebra of $\mathbb{R}$-valued continuous functions on $X$. In partic …

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed i …

Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow. Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative …

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the …

I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the …

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was …

Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds. $L^2$-version. Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \ …

If you read through the paper, Borel and Remmert briefly describe their proof strategy: They show that every compact homogeneous complex manifold $V$ can always be fibered in two different ways: $V \ …

To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$. To (Q2): Yes, see (Q4). …