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I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1: Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a bundle …

I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-1}(M, \mat …

I know it is easy to classify line bundles over $\mathbb{R}P^2$. But do we have a classification of two dimensional complex vector bundles over $\mathbb{R}P^2$?

Let $E$ be an $n$-dimensional vector bundle on a manifold $M$ and $\nabla: \Gamma(E)\to \Omega^1(M,E)$ be a flat connection on $E$. Classical Riemann-Hilbert correspondence tells us that ker$\nabla$ i …

Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact. Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $ …

I'm not sure whether the level of this question is suitable for Mathoverflow. Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow …