For questions on vector bundles.

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2
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1answer
27 views

Is a bundle morphism which restricts to homeomorphisms of the fibers a bundle isomorphism?

If $f$ is such a map between total spaces (assume a common base space) then it is a bijective and continuous and the inverse will be fiber preserving so that all we would need to prove is that the ...
2
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0answers
29 views

Cylinder as trivial fiber bundle with fiber $S^1$?

In prepartion for the example of a Mobius strip, a cylinder is often taken as a first example of a trivial fiber bundle with fiber $I$ and total space $S^1\times I$. However, it seems to me that we ...
2
votes
1answer
70 views

Splitness of quotient sequence

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A', B', C'$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B ...
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0answers
24 views

Showing that Killing vector fields form a vector space without introducing connection

I'm trying to show that the sum of two killing vector fields is again a vector field without introducing a connection and just using the smooth structure on a manifold. Let $X,Y$ be Killing vector ...
1
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0answers
44 views

Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
2
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0answers
23 views

Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
2
votes
1answer
30 views

Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} ...
1
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0answers
31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
1
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0answers
22 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
2
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0answers
52 views

Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and ...
1
vote
1answer
28 views

Definition of Levi-Civita connection map?

Does anyone know definition of Levi-Civita connection map that defined as $TTM\to TM$. I would appreciate if you could give a good reference. Thanks.
2
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1answer
32 views

How many distinct flat connections are there on a flat bundle?

Given a flat smooth vector bundle (i.e. with constant transition functions), how many distinct flat connections could we put on it? If the flat connection is not unique, is it unique up to gauge ...
4
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1answer
62 views

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 ...
2
votes
1answer
55 views

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 ...
2
votes
1answer
22 views

Chern character of canonical line bundle over $\mathbb{CP}$

Let $H \to \mathbb{CP}$ be the canonical line bundle over $\mathbb{CP}=S^2$. Then from the text Vector Bundles and K-theory by Hatcher, given the chern character, $ch$, and first chern class ...
7
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3answers
1k views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...
0
votes
1answer
24 views

Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
4
votes
2answers
821 views

Understanding the Definition of a Differential Form of Degree $k$

Let $M$ be a smooth manifold. A differential form of degree $k$ is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. Does it mean that a differential form of degree ...
10
votes
2answers
269 views

Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
0
votes
1answer
9 views

Morphisms and Whitney Sum of bundles

Suppose that $X$ is a topological space and $(E,p_1)$ and $(F, p_2)$ are locally trivial vector bundles over $X$. Consider bundle $E \oplus F$. Is it true that $i_E$ defined by $E_x \to (E \oplus ...
1
vote
1answer
26 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 ...
3
votes
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 ...
2
votes
2answers
50 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 ...
3
votes
1answer
52 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 ...
1
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0answers
28 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 $A$ ...
0
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0answers
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 ...
1
vote
0answers
15 views

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 ...
1
vote
1answer
40 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?
0
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0answers
55 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 ...
0
votes
1answer
24 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 $S^2$ ...
-1
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1answer
27 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 ...
4
votes
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 ...
1
vote
0answers
22 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 ...
1
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0answers
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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0answers
26 views

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 ...
0
votes
1answer
25 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 $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
0
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0answers
21 views

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 $N(Z;Y)$ ...
0
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0answers
20 views

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 ...
0
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0answers
37 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 ...
2
votes
1answer
57 views

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?
5
votes
1answer
43 views

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 ...
2
votes
0answers
15 views

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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0answers
20 views

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) ...
0
votes
1answer
30 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 ...
1
vote
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 ...
0
votes
1answer
33 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 ...
0
votes
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 ...
0
votes
0answers
5 views

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 $T(I\times I)\longrightarrow A$? ...
0
votes
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 ...
1
vote
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 ...