For questions on vector bundles.

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Equivalent condition for continuity of a bundle map

Let $U$ be a topological space, and let $U \times \mathbb{R}^n$ be the trivial bundle. Let $\varphi: U \times \mathbb{R}^n \to U \times \mathbb{R}^n$ be a bundle map, but assume we don't know whether ...
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1answer
21 views

Find a nontrivial bundle of $S^1$ with fibre isomorphic to $\mathbb{R}^n$

Show that such a nontrivial bundle exists for every $n\in\mathbb{N}$. I don't really have any useful ideas here. I'm not sure if there is a general approach I should be taking or if there is just a ...
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1answer
66 views

Trivialised sub-bundle of a trivialised bundle and its orthogonal bundle

I'm reading Kirby's The Topology of $4$ Manifolds, and I encountered the following claim: Recall that a trivialised sub $k$-plane bundle of a trivialised $(n+k)$-plane bundle determines a ...
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2answers
44 views

Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
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1answer
46 views

Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
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26 views

the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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15 views

Which constructions on vector bundles satisfy a universal property?

I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property. What I mean by "some kind of universal ...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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1answer
33 views

Euler classes of oriented $2$-dimensional vector bundle, oriented $S^1$-bundle same?

As the question title suggests, are the Euler classes of an oriented $2$-dimensional vector bundle and of an oriented $S^1$-bundle the same?
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49 views

Quick question: Line bundle on union of two lines

Let $l_1$, $l_2$ be two lines in $\mathbb{P}^n$. What is the meaning of $\mathcal{O}_{l_1}(a_1)\cup\mathcal{O}_{l_2}(a_2)$ as a sheaf on the union $C=l_1+l_2$ of two distinct lines and why do we ...
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1answer
22 views

Non-existence of $1$-dimensional tangent distributions on the sphere $S^2$

I am reading about geometric quantization and real polarizations, and it is claimed that there exist no real polarizations on the sphere $S^2 \subset \Bbb R^3$ because every tangent field on $S^2$ ...
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1answer
21 views

Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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1answer
35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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20 views

Why is $H_{DR}^p(M,\mathbb{C})\cong H_{DR}^p(M,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}$

This question is related to my previous question. The answers to that question inspired a new question, namely For a complex manifold $M$, why is $H_{DR}^p(M,\mathbb{C})\cong ...
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27 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...
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1answer
24 views

Bundle that is isomorphic to the bundle of Whitney sum

I am involved with one question that a friend of mine asked. If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
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Normal bundle of sub-manifold is a manifold

This is exercise 2.3.12 of Guillemin and Pollack: Let $Z$ be a sub-manifold of $Y$, where $Y \subset \mathbb{R}^M$. Define $N(Z;Y)=\{(z,v):z\in Z, v\in T_z(Y), v \perp T_z(Z)\}$. Prove that $N(Z;Y)$ ...
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Topology of bundle maps in Atiyah-Singer IV

I'm trying to read "The index of elliptic operators IV" by Atiyah and Singer, and I do not understand why the topology on $\mathrm{Diff}(X,E)$ on page 123 is constructed in such a peculiar way. Is ...
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36 views

For vector bundles $A\to E$, $B\to E$, is $\Gamma(A\oplus B)\cong \Gamma(A)\oplus\Gamma(B)$?

I've wondered for a while now if for vector bundles $A\to E$, $B\to E$, is $\Gamma(A\oplus B)\cong \Gamma(A)\oplus\Gamma(B)$? I found a related question here, on this question for tensor products, but ...
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1answer
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Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
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Chern character in odd K-theory

I'm familiar with the definition of Chern character for a vector bundle. This leads to the definition of Chern character for $K$ theory (even theory) with values in even cohomology (the definition ...
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Compact Vertical Cohomology and Euler Class of CP1

First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently ...
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Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
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1answer
28 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
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2answers
63 views

How to find two inequivalent ,but weakly equivalent bundles?

I want to know if I can have an inequivalent but weakly equivalent bundles over the Torus, i.e. if we can find explicitly an example of a bundle map $(\overline{f},f)$ where we require that ...
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1answer
28 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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1answer
16 views

Cocycle condition on bundles as some equivalence relation?

The three cocycle conditions on the transition maps of a fiber bundle look alot like the three condition defining equivalence relation - refelxivity, symmetry, and transitivity. I was wondering ...
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Vector bundle morphisms $T(I\times I)\longrightarrow A$?

Let $I:=[0, 1]$ be the unit interval in $\mathbb R$ and $\pi:A\longrightarrow M$ a vector bundle. Is there a nice characterization of the vector bundle morphisms $T(I\times I)\longrightarrow A$? ...
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1answer
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Normal bundle associated to $\mathbb{R}P^n\hookrightarrow \mathbb{R}P^{n+1}$ is the tautological line bundle

I'm interested in proving the fact of the title and I was following the reasoning in page 8 here (at the end). There is a step which are totally unclear to me, namely the identification of the normal ...
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1answer
18 views

The square root of this operator is an isometry?

This question is motivated by the following proposition from the book "Connections, curvature and cohomology" by Werner Greub: If $(\xi, g)$, $(\eta, h)$ are riemannian vector bundles over the ...
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1answer
17 views

Vector Bundles Morphism Properties?

Let $p_A:A\longrightarrow M$ and $p_B:B\longrightarrow N$ be vector bundles and $\Phi:A\longrightarrow B$ a vector bundle morphism covering $\phi:M\longrightarrow N$. For $p\in M$, define ...
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1answer
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Is it true that $A_p=\{\alpha(p): \alpha\in \Gamma(A)\}$ for a vector bundle $A\longrightarrow M$?

Is it true that if $A\longrightarrow M$ is a vector bundle then $A_p=\{\alpha(p): \alpha\in \Gamma(A)\}$ for every $p\in M$? Here $A_p$ is the fiber over $p$ and $\Gamma(A)$ are the smooth sections ...
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1answer
28 views

On the Euler Class of a manifold

Let $M$ be an orientable, compact $n$ dimensional differentiable manifold and $e \in H^n(M, \mathbb{Z})$ the Euler class of the tangent bundle of $M$, defined via the Thom Isomorphism. Also, let $[M] ...
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$L^2$ space with values in vector bundle.

Let $E$ be a vector bundle over $S^2$ with inner product, $\pi$ its bundle projection and each fiber has dimension $m$. Let $\Omega \subset \mathbb{R^n}$. Consider space of all square integrable ...
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1answer
64 views

Sobolev spaces of sections of vector bundles

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset ...
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Pull back and intersection of divisors

Let $X$ be a smooth projective variety over complex numbers. Let $Z$ be a smooth closed subscheme of $X$. Let $L$ be a very ample line bundle on $X$. Then $L|_Z=E$ is a very ample line bundle on $Z$. ...
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1answer
41 views

Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
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1answer
83 views

Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
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1answer
45 views

Classification of rank $\geq 2$ vector bundles over Grassmannians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmannian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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quick question about ampleness of the following line bundle.

I have a quick question concerning the ampleness of the following line bundle, it came up in an article I've been trying to read. Suppose that $\pi: Y \rightarrow X$ is a morphism of complex ...
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If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
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1answer
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Confused by two different perspectives on $G$-vector bundles

I'm trying to understand how these two perspectives on vector bundle with a $G$-action come together. Perspective 1: Let $P \to X$ be a principal $G$-bundle. The associated bundle construction gives ...
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1answer
76 views

Is $C^\infty (D^n,S^n)$ compact?

Is $ C^\infty (D^n,S^n)$ compact where $D^n$ is the unit closed disc in $\mathbb{R^n}$? I got this question when reading a paper. The author mentioned without explanation that $\{ \alpha \in \Gamma ...
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14 views

Degree of line bundles with a non-trivial morphism between them

We assume that $X$ is a K\"ahler manifold. $L_1$ is a holomorphic line bundle and $F$ is a subsheaf of rank $1$. If there is a non-trivial morphism $F\to L_1$, then $\operatorname{deg}F\leq ...
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1answer
29 views

Non-zero tangent vectors in Hermitian manifolds

I'm a physicist trying to learn about Chern's class and forms by myself. To that respect I feel the Chern's writings far more pleasant than esoteric literature written by and for physicists, which are ...
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1answer
69 views

Relationship Between Connections on a Vector Bundle and a Riemannian Base

I'm starting to get acquainted with how to define affine connections on a vector bundle. Suppose $\pi: E \to M$ is a rank $k$ vector bundle over a Riemannian manifold $M$ with metric $g$, where we ...
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1answer
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Two vector bundles over same base manifold $X$

What are two vector bundles over the same base manifold $X$ which are isomorphic as vector bundles in the general sense, but not isomorphic over $X$? (That is to say, this would demonstrate that there ...
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1answer
33 views

Restrictions of $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if X is paracompact

The following assertion appears in Milnor's Characteristic Classes. The restrictions of a vector bundle $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if $X$ is paracompact. ...
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1answer
36 views

Are two vector bundles Möbius band and $S^1\times \mathbb{R}$ isomorphic as vector bundles over $S^1?$

If $\forall (x,t), (y,t^{\prime}) \in \mathbb{R}\times \mathbb{R}$ define $$(x,t)\backsimeq(y,t^{\prime})\Leftrightarrow \exists n\in \mathbb{Z} ; x=y+2n\pi , t=(-1)^{n} t^{\prime} $$ $$x\backsim y ...