1
vote
0answers
21 views

Integration of the components of a vector field along a curve

Let $M$ be a smooth manifold, $\gamma:[0,\tau]\rightarrow M$ a smooth curve and $X$ a vector field which does not vanish on $\gamma$ and is not tangent to $\gamma$. On $M$ we consider a vector bundle ...
6
votes
1answer
37 views

Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?

Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts ...
0
votes
1answer
35 views

Unitary connection and Hermitian connection

I am confused with the two notions. I basically understand Hermitian connection: it is a complex analog of metric-compatible connection on $TM$, a connection that preserves the hermitian metric $h$ ...
0
votes
1answer
34 views

The tangent bundle over a manifold is trivial iff the manifold is paralelizable

Why is the tangent bundle over a manifold trivial if and only if the manifold is parallelizable? What additional condition do we need to impose on a fiber bundle if we want it to be trivial exactly ...
0
votes
0answers
18 views

The tangent bundle over a manifold is locally trivial

Assume $M$ is an $m$-dimensional manifold and $(U,h)$ is a chart. Denote the differential of $h$ at the point $p$ by $T_ph$. How can I verify that $pr_1(Th(p,x))=h(p)$ where $pr_1$ is the projection ...
4
votes
1answer
55 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
0
votes
1answer
25 views

Vanishing of non top-order Chern classes

Let $E \to B$ be a rank-$r$ complex vector bundle and denote by $c_1(E)$, $\ldots$, $c_r(E)$ its Chern classes. Then $c_r(E)$ is just the Euler class of the realization of $E$ as a real vector bundle ...
2
votes
1answer
51 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
1
vote
3answers
64 views

First steps with Vector Bundles

I've concluded a differential geometry course (mainly covering classical results about parametric surfaces or diff. surfaces in $\mathbb{R}^3$, for example Gauss' Theorema Egregium or geodetics) which ...
3
votes
1answer
38 views

Boundedness of the operator $[\Lambda, \Theta]$.

I am reading Griffiths-Harris book on algebraic geometry and in "Theorem B", where he proves (the analogue of Serre's vanishing theorem) that Let $M$ be a compact, complex manifold, $L\rightarrow ...
1
vote
1answer
34 views

A connection is the limit of the newton quotient of the parallel transport

Let $E\rightarrow M$ be a vector bundle with connection $\nabla$. Denote by $\Pi_{\gamma(t_{0})}^{\gamma(t_{1})}:E_{\gamma(t_{0})}\rightarrow E_{\gamma(t_{1})}$ the parallel transport map along the ...
2
votes
1answer
42 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
3
votes
1answer
34 views

Why is the restricted holonomy the identity component of the holonomy group?

Let $M$ be a connected smooth paracompact manifold, $E$ a vector bundle over $M$ with fibre $\mathbb R^k$, and $\nabla$ a connection on $E$. It is known that Hol$^0(\nabla)$ is a connected Lie ...
1
vote
1answer
64 views

How a principal bundle and the associated vector bundle determine each other

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...
1
vote
0answers
20 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
1
vote
1answer
52 views

Why is the space of all connection on a vector bundle an affine space?

I think this result is very well known, but I don't understand its proof. Let E a vector bundle over a manifold M, and $\Omega^i(E):=\Gamma(\Lambda^iT^*M\otimes E)$ the space of E-valued differential ...
-4
votes
1answer
269 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
3
votes
1answer
46 views

A connection over a 1-dim manifold is flat

Let $M$ be a 1-dimensional manifold and let $E$ be its vector bundle. I want to show that every connection $D$ on this vector bundle is flat. A connection $D$ is flat means that we have $$D_v D_w ...
1
vote
2answers
55 views

Is not the surjective map $\pi$ associated with a vector bundle infact a bijection?

I am reading John M Lee's Riemannian Manifolds : An Introduction to Curvature, which is very well written. On page 16 : "Vector bundles are defined", quoting A (smooth) $k$-dimensional vector ...
0
votes
0answers
34 views

Finding frame bundles

Let $P\to M$ be a vector bundle of manifold $M$ with finite rank $n$. Is there any method to finding frame bundle $Fr(P)$ on $Fr(P)\to M$.
2
votes
1answer
91 views

Normal bundle of a hyperplane section

Let $Y\subset \mathbb{P}^n$ be a smooth projective variety and let $H$ be a smooth hypersurface in $\mathbb{P}^n$ such that $Z=Y\cap H$ is smooth. How are the normal bundles of the various embeddings ...
2
votes
1answer
43 views

On the structure of a vector bundle

Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and ...
1
vote
1answer
36 views

Action of a Lie group on the tangent bundle..

Let $P\longrightarrow M$ be a $G$-principal bundles. How do I define an action of $G$ over $TP$? Furthermore how can I identify the space of sections $\Gamma(TP/G)$ with $\mathfrak{X}(M)^G$ where ...
2
votes
0answers
31 views

Restriction of a Lie bracket on the space of section of a vector bundle..

Let $A\longrightarrow M$ be a vector bundle and $U\subseteq M$ an open set. Suppose I have a lie bracket on $\Gamma(A)$ such that if $\rho:A\longrightarrow TM$ is a bundle map then $$[a, fb]=f[a, ...
0
votes
1answer
23 views

What would be the space of section of the bundle $\mathfrak{g}\longrightarrow \{e\}$?

Let $\mathfrak{g}$ be a Lie algebra and $\pi:\mathfrak{g}\longrightarrow \{e\}$a vector bundle over a point. What would be the sections of this bundle?
1
vote
1answer
40 views

a question about compact tangent bundle

I have a question about tangent bundles. Is there a compact tangent bundle? Or what conditions do we need to be sure that tangent bundle of a manifold be compact?
2
votes
2answers
39 views

Tangent bundle of sphere with $g$ handles

How can one show that tangent bundle $TM$ is not trivial if $M$ is a sphere with $g$ handles and $g \ne 1$?
1
vote
0answers
73 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
1answer
73 views

Connections and covariant derivatives

Let $A$ be a connection on a principal $G$-bundle $P$, let $\chi :G\rightarrow GL(V)$ be a representation of $G$, and let $E:=P\times _\chi V$ be the associated gauge bundle. Then, there is a ...
1
vote
1answer
38 views

Maps of $G$-bundles

A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by ...
7
votes
1answer
85 views

Line bundles over $\mathbb R P^2$

As in this post, I'm continuing studying line bundles. Now it's line bundle over $\mathbb R P^2$. I know that this bundle is not trivial. So I want list up to equivalence all bundles over $\mathbb R ...
3
votes
2answers
97 views

Line bundle over $S^2$

I'm trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I'm interesting in another approach. For the sphere there is two maps : upper ...
1
vote
1answer
82 views

How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...
2
votes
1answer
75 views

a question about space of smooth sections

Let $\Gamma(M,L) $ be the space of smooth sections, then why $\Gamma(M,L) $ is isomorphic to $A=\{f:L^{\times}\to \mathbb{C}; f(cz)=c^{-1}f(z), c\in \mathbb{C}-\{0\} , z\in L^{\times}\}$ . Here ...
1
vote
0answers
27 views

Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
2
votes
1answer
28 views

If $dX_1 = dX_2$ then curvatures of $\nabla^{X_1}$ and $\nabla^{X_2}$ agree

Let $E \simeq M \times \mathbb C$ be a trivial smooth complex line bundle over the Riemann surface $M$ and let $S \colon M \to E$ be its smooth nowhere vanishing section. Let $\nabla^1$ and $\nabla^2$ ...
2
votes
0answers
39 views

Partial Differential Operators on Vector Bundles

can anyone suggest me a nice reference for partial differential operators on vector bundles? Thanks..
0
votes
1answer
57 views

Cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold

How to prove that cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold? Detailed explanation is welcome. Thanks in advance.
0
votes
1answer
27 views

Which set is this $I_p(p)\cdot \Gamma(E)$?

Let $\pi:E\rightarrow M$ be a smooth vector bundle and $p\in M$. Consider $$I_p(M)=\{f\in C^\infty(M): f(p)=0\},$$ and $\Gamma(E)$ the $C^\infty(M)$-module of smooth sections over $E$. Notice $I_p(M)$ ...
6
votes
1answer
87 views

Every vector bundle over $[0,1]^n$ is trivial

I would like to show the followoing result: Every vector bundle over $[0,1]^n$ is trivial First, I consider the case $n=1$, so let $E$ be a vector bundle over $[0,1]$. If $\nabla$ is a connexion ...
6
votes
2answers
82 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
1
vote
1answer
45 views

is normal bundle of a manifold trivial?

If you embed a manifold $M$ in Euclidean space, is the normal bundle always trivial? Or give an example with non-trivial normal bundle.
2
votes
2answers
67 views

How to show the existence and uniqueness of the pullback connection in vector bundles?

There is the following result: If $D$ is a connection on a vector bundle $E$ over $N$ and $φ$ is a smooth map from $M$ to $N$, then there is a pullback connection on $φ^*E$ over M, determined ...
6
votes
4answers
188 views

Line bundles of the circle

Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious ...
0
votes
0answers
47 views

Sections on the Tautological Line bundle $E(\gamma_n)$..

I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb ...
2
votes
2answers
54 views

Problem with normal bundle to sphere $\mathbb S^n\subset \mathbb R^{n+1}$

I'm in trouble for understanding the normal bundle to $\mathbb S^n\subset\mathbb R^{n+1}$. By definition $$\nu(\mathbb S^n)=\bigcup_{p\in\mathbb S^n}\nu_p(\mathbb S^n),$$ where $\nu_p(\mathbb ...
1
vote
1answer
83 views

Normal Bundle is Trivial

how can I show the normal bundle $$\nu(\mathbb S^n):=\bigcup_{p\in\mathbb S^n} T_p\mathbb R^{n+1}/T_p\mathbb S^n, $$ is a trivial bundle? Any help will be valuable.. Thanks
4
votes
1answer
53 views

How to show $f^*E$ is a smooth submanifold…

I'm wondering how to show the following: let $E$, $B_1$ and $B_2$ smooth manifolds. Suppose $\rho:E\rightarrow B_2$ is a smooth vector bundle and $f:B_1\rightarrow B_2$ a smooth map. If we write ...
2
votes
1answer
78 views

$SO(3)$ diffeomorphiс to a spherization of tangent bundle of two-dimensional sphere?

Let $\xi:Y\to X$ be a vector bundle. Spherization of this bundle is the quotient space $(X\backslash \Theta) / \sim $ (here $\Theta$ is a graph of the zero section of the bundle) with equivalence ...
6
votes
2answers
170 views

Vector bundles of rank $k$ with base $S_{1}$

I don't understand how to solve the following problem. Can you help me? Prove that there are only two vector bundles of rank $k$ with base $S^{1}$ $-$ trivial $1_{k}$ and non-trivial $\eta_{k}$. ...