Questions tagged [vector-bundles]

Question 1

Let $M$ and $N$ be two smooth manifolds. Let $f:M \to N$ a smooth map. Let $E \to N$ be a vector bundle with orientation $o$, where I define the latter as a choice of (global) section $o: N \to \...

Question 2

Let $V\to X$ be a real vector bundle with structure group $SO(3)$. Why does the second Stiefel-Whitney class satisfies $$ w_2(V)^2=p_1(V) \mod 4?$$ This is asserted in p.41 of Donaldson & ...

Question 3

Let $M$ be smooth manifold and $\pi:E\to M$ a vector bundle. I am working on a smooth version of the classification of vector bundles. For this, I need to show that the map $$\Phi:[M,\operatorname{Gr}...

Question 4

I was studying the Thom space construction of vector bundles. As a simple case, I tried to understand the Thom space of real, rank one, trivial vector bundle ($E$) over $S^1$. According to the ...

Question 5

I am interested in learning more about general vector bundle theory. More specifically, vector bundles of class $C^k$ for $k\in\mathbb{N}$ or $C^\infty$ or real-analytic whose fibers can be given the ...

Question 6

Let $E$ be a rank 2 complex vector bundle over a 4-dimensional manifold $X$. (I believe that the argument below does not depend on the rank of $E$ and the dimension of $X$.) I want to try to show, for ...

Question 7

Suppose that $E_0, E_1 \rightarrow M$ are two $k$-dimensional vector bundles over a manifold $M$ classified by maps $\phi_0, \phi_1: M \rightarrow BGL(k)$. If $\phi_0, \phi_1$ are homotopic, then $E_0,...

Question 8

I am superficially quoting the following result from Algebraic Geometry by Hartshorne. I was told the following from a brief conversion with a graduate student, and became very interested. Please be ...

Question 9

I'm looking for a pair of smooth Vector bundles with common base space, in which the total spaces are homeomorphic but non-diffeomorphic smooth manifolds. I was trying to construct trivial bundles ...

Question 10

Let $M$ be a $n$-dimensional Riemannian manifold, i.e. $M = \cup_{\lambda \in \Lambda}\, (U_{\lambda}, g_{\lambda})$ with each $U_{\lambda}$ being an open ball around the origin of ${\Bbb R}^n$ and $...

Question 11

I'm trying to follow my lecture course in fiber bundles and I'm not sure whether my lecturer has made a mistake. Definition. Let $M$ be a complex manifold. Then define the picard group $\mathrm{Pic} (...

Question 12

Let $M$ be a Riemannian manifold, $E\to M$ be a vector bundle with bundle metric, and equip $E$ with a metric-compatible connection $\nabla^E$, denote the curvature tensor of $\nabla^E$ by $R^E$. In a ...

Question 13

Suppose X be a Riemann surface endowed with anti-holomorphic involution $\sigma_X$. $V$ is a holomorphic vector bundle on $X$ with holomorphic connection $D$. It is a fact that $\sigma_X^*\overline{V}$...

Question 14

Let $P_{\Bbb C}^1$ be a projective line over the complex number field ${\Bbb C}$. We have $P_{\Bbb C}^1 \thickapprox S_{\Bbb R}^2$ as the Riemannian manifolds. There is the Levi-Civita connection on $...

Question 15

In singularity theory, one defines an intrinsic derivative for a vector bunle homomorphism $\phi: E \rightarrow F$ where $E\xrightarrow{\pi_E}B$ and $F\xrightarrow{\pi_F}B$ are vector bundles with ...

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Questions tagged [vector-bundles]

(When) do orientation sections pull back?

$w_2(V)^2=p_1(V) \mod 4$ for $SO(3)$ bundles

“Smooth” proof for vector bundle classification

Thom space of a rank one, trivial real vector bundle over $S^1$

Book Recommendation for Vector Bundles

Showing that the trace of the difference of the square of the curvatures of two connections on a vector bundle is exact

Vector bundles with equivalent sphere bundles

Conversion between Cartier Divisors, Weil Divisors, Line Bundles and Invertible Sheaves.

Homeomorphic smooth Vector bundle that aren't diffeomorphic

Why is the connection necessary?

First Chern class of a line bundle

Describing sections in the kernel of curvature tensor

Induced connection on pull-back of conjugate bundle

Line bundle and the ‘‘algebraic’’ Levi-Civita connection.

Intrinsic derivative

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