For questions on vector bundles.

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Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$?

Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$? I know that ...
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1answer
19 views

Is the tangent bundle of an oriented surface plus a trivial bundle trivial?

Let $\Sigma$ be an oriented closed surface and $E$ be the direct sum of $T\Sigma$ with a trivial line bundle. Is $E$ a trivial rank $3$ vector bundle? For genus $0$ and $1$ the answer is yes since ...
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2answers
24 views

iff criterion for n-dimensional vector bundle being trivial

Prove that an $n$ dimensional vector bundle $p: E \to B$ is trivial iff there exists a continuous map $\pi: E \to \mathbb{R}^n$ whose restriction to each fiber $p^{-1}(b), b \in B,$ is a vector space ...
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1answer
88 views

Working out an example in Hatcher vol. $2$: Pullbacks of the Möbius Bundle

I'm working out the examples made by Hatcher to shows some pullbacks (definition here for clarity) and this ("simplified" version with $n=2$ or $n=3$) gave me an hard time: $$\times \times \times ...
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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 ...
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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 ...
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1answer
96 views

What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
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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 $h$ ...
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1answer
141 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
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1answer
91 views

Exactness of the pullback of the Euler sequence

Let $(L,V)$ be a base point free $g^r_d$ on a curve $C$. Then we have an exact sequence $0\rightarrow M_{L,V}\rightarrow\mathcal{O}_C^{r+1}\xrightarrow{\theta} L\rightarrow 0$, where we just let ...
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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 ...
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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 ...
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34 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
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1answer
173 views

When is the pushforward / direct image of a reflexive sheaf locally free?

I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again. This makes me ...
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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. ...
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1answer
19 views

Stiefel-Whitney Classes: Simple Example

I need help finding the Stiefel-Whitney classes $w_k(\eta)$ of the normal bundle of the $n$-sphere. Now since $H^k (S^n ; \mathbb{Z}/2\mathbb{Z}) =0$ for $k \neq 0,n$, then $w_k(\eta) =0$ for $k ...
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1answer
26 views

Universal Equivariant Line Bundles

Complex line bundles are classified by maps into the universal bundle $\gamma\rightarrow \mathbb{C}P^\infty$. If I wanted to talk about $G$-equivariant line bundles over $X$, is there a corresponding ...
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1answer
28 views

Trivialization of the normal bundle of a knot

Let $ \phi $ be an embedding of $S^1$ in $ R^3$ or $S^3$. It is often mentionned (for instance when discussing framed knots) that one can choose a trivialization of the normal bundle $ \nu ...
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1answer
20 views

Strictly convex norms instead of inner products on the tangent spaces

On differentiable manifolds, is there a term for functions $f$ from the tangent bundle to the real line such that for all points of the manifold, the restriction of $f$ to that point's tangent space ...
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0answers
15 views

Calculus of final point of a rect/line

I'm creating a little game and i'm trying to calculate the final point to draw the bullet movement. I getting troubles trying to calculate the final point of the line having the bullet vector [-1,0 ...
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1answer
25 views

Are vector bundles isomorphic when their transition functions are homotopic?

I'm in the process of trying to understand the equivalence between two different approaches to vector bundles, namely defining them to be smooth surjections $E\to B$ that are locally isomorphic to ...
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35 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
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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 ...
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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 ...
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Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
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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 ...
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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 ...
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1answer
63 views

Dual of a Vector Bundle

I am not quite getting the idea of morphisms between vector bundles. I read and reread the definition but I didn't quite get it. Can someone provide me with an example of a morphism between a vector ...
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1answer
26 views

Necessity - Reasons of a passage in a proof of Hatcher vol. 2

Some words about the context of the proposition. Hatcher is defining operations on vector bundles over the same base space $B$. We are speaking about the Whitney sum here, and it is proving that if ...
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2answers
60 views

Surjective map from a trivial bundle to any vector bundle

I was reading over some notes on vector bundles which make use of the following fact: If $X$ is a $n$-manifold and $V$ is a real vector bundle on $X$ of rank $k$, then there exists a surjective ...
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69 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
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1answer
46 views

Measurable vector bundles trivial

I hope you can help: If $E$ is a measurable vector bundle over a compact metric space $(X,\mu)$ then there is a subset $Y\subset X$ such that $\mu(Y)=1$ and $\pi ^{-1}(Y)$ is isomorphic to a trivial ...
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1answer
94 views

Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , ...
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0answers
50 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
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3answers
65 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 ...
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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 ...
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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 ...
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36 views

Vector bundle on a transversal intersection

Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such ...
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31 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
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1answer
188 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle ...
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0answers
89 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
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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 ...
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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 ...
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1answer
52 views

Section of a vector bundle as a submanifold

I am currently working on one part of a problem surrounding sections and submanifolds. Given a real vector bundle $\pi: E\rightarrow M$ of rank k, with a smooth global section $s:M\rightarrow E$, can ...
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Exterior algebra as a complex of vector bundles and an element in relative K-theory

There is a description of relative $K$-theory in terms of complexes of vector bundles. The following is an excerpt from Atiyah's book. Let $V$ be a complex vector space and consider the exterior ...
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1answer
77 views

Canonical line bundle over a projective bundle

The following is an excerpt from the Atiyah's K-Theory. If $E$ is a vector bundle over $X$ then each point $a\in P(E)_{x}=P(E_{x})$ represents a one-dimensional subspace $H_{x}^{*}\subset E_{x}$. ...
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1answer
28 views

Projective bundle associated to sum of trivial line bundles

I am having some difficulty seeing that $P(1\oplus 1)=X\times S^{2}$. Here $1$ denotes the trivial (complex) line bundle over $X$, and $P$ indicates the projective bundle.
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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 ...
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1answer
55 views

Krull-Schmidt-Remak for vector bundles

I'm reading Nori's paper The fundamental group scheme, and I have some problems in certain passages of the proofs. This one is from chapter 1, 2.3. Let $X$ be a complete connected reduced ...
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1answer
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Ampleness, Nakai's criterion and pullback

In the book I'm reading ( Geometry of Algebraic Curves ), at some point (page $310$) they make the following claim: One can use Nakai's criterion to establish the general fact that if $f:X\to Y$ ...