2
votes
1answer
35 views

computing transition function of tangent bundle $S^n$

I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$. I started with the stereographic atlas $U_1 = S^n - \{N\}$ and $U_2 = S^n - \{S\}$ ...
4
votes
0answers
35 views

Non-vanishing differential forms and flatness

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
0
votes
1answer
40 views

Tangent bundle of a tubular neighborhood

Let $N \to X$ be normal bundle of a submanifold $X$ of $Y$. How can I prove that $TN|_{TX}$ is isomorphic to the normal bundle of the inclusion $TX\to TY$? And why this vector bundle is isomorphic to ...
3
votes
1answer
68 views

Can we measure how close a vector bundle is to being trivial?

For a vector bundle $E$, I will denote the maximum number of linearly independent global sections of $E$ by $\eta(E)$. We have $\eta(E) \in \{0, 1, \dots, \operatorname{rank}(E)\}$ and $\eta(E) = ...
4
votes
2answers
101 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
1
vote
2answers
32 views

Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...
1
vote
0answers
22 views

Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
1
vote
1answer
108 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
3
votes
0answers
49 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
1
vote
1answer
93 views

R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
1
vote
0answers
36 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
0
votes
3answers
51 views

The difference between a fiber and a section of a vector bundle

If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then ...
2
votes
0answers
35 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
3
votes
1answer
65 views

Are vector bundles just modules over $C^{\infty}(M)$, or are “locality” conditions required?

In this question the asker defines 1-forms on a (real, smooth) manifold $M$ to be $C^{\infty}(M)$-module homomorphism[s] from $Vec(M)$ to $C^{\infty}(M).\:\:\:\:(*)$ I'm wondering if this is ...
4
votes
0answers
50 views

Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
1
vote
1answer
48 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
2
votes
0answers
64 views

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
2
votes
0answers
48 views

Prove the existence (or well-definedness) of the induced connection in tensor bundle

Given a connection $\nabla$ on a vector bundle $E$ over a smooth manifold $M$, we know there is a unique extension of $\nabla$ to all tensor bundles of $E$ that satisfies Leibniz rule and contraction. ...
0
votes
1answer
44 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
0
votes
0answers
42 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
2
votes
1answer
77 views

Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
0
votes
1answer
43 views

Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...
2
votes
2answers
105 views

Recover Covariant Derivative from Parallel Transport

It is well known that one can recover the connection from the parallel transport. I struggle to understand this concept. Since $\Gamma(\gamma)^t_s:E_{\gamma(s)}\to E_{\gamma(t)}$ is an isomorphism ...
1
vote
1answer
32 views

Tensorproduct of vectorbundles

Assume we got $\pi:E \to M$, a vector bundle of manifolds. I know how to make the bundle $(E \otimes E)^* \to M$. But how do the local trivializations look like ? I suppose that if ...
6
votes
1answer
76 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
115 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
56 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
22 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
83 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
3answers
51 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
58 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
74 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
40 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 ...
3
votes
1answer
71 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
44 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 ...
2
votes
1answer
136 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 ...
2
votes
1answer
90 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
299 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
55 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
62 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
38 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
108 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
46 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
54 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
38 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
24 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
45 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
44 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 $g \ne 1$?