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Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

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33 votes
1 answer
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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 ...
Keenan Kidwell's user avatar
2 votes
1 answer
1k views

Global Trivialization of $M\oplus M$

$\mathbb S^1$'s $\mathbb R^1$-bundle is $$\{\mathbb S^1\times\mathbb R^1\text{, open Möbius strip}\}$$ and its $\mathbb R^2$-bundle is $$\{\mathbb S^1\times\mathbb R^2\text{, open solid Klein bottle}...
gaoxinge's user avatar
  • 4,504
71 votes
2 answers
5k views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
Martin Brandenburg's user avatar
36 votes
4 answers
14k views

Why is the tangent bundle orientable?

Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?
Jr.'s user avatar
  • 4,056
26 votes
3 answers
5k views

Vector bundle transitions and Čech cohomology

I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ (...
Benjamin's user avatar
  • 2,816
14 votes
2 answers
4k views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
Joachim's user avatar
  • 5,345
12 votes
1 answer
1k views

Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?

On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$. Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf A^1$...
Brenin's user avatar
  • 14.1k
8 votes
2 answers
661 views

Defining Connection in Hom Vector Bundle without Coordinates, using Dieudonne's Definition

In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given: A linear connection $C$, in a vector bundle $(E,\pi, M)$ ...
peek-a-boo's user avatar
  • 57.4k
47 votes
5 answers
4k views

Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
Zev Chonoles's user avatar
33 votes
1 answer
7k views

Tangent Bundle of Product Manifold

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that: $T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $ Can we conclude that $T(M\times N) \cong T(M) \...
Mark B's user avatar
  • 1,994
21 votes
3 answers
5k 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 ...
Seirios's user avatar
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20 votes
1 answer
2k 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 ...
Dominik's user avatar
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14 votes
1 answer
3k views

The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
caffeinemachine's user avatar
9 votes
1 answer
1k views

Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial

Let $X$ be a projective scheme over a field $k$. Let $\mathcal{O}(1)$ be an ample line bundle on $X$, then the Hilbert polynomial $P(E)$ is given by $m\mapsto\chi(E ⊗ O(m))$. The explicit polynomial ...
gradstudent's user avatar
  • 3,372
8 votes
2 answers
628 views

Why is the tangent bundle defined using a disjoint union?

In textbooks about differential geometry, one finds often the disjoint union in the definition of the tangent bundle (e.g. in "Lee: Introduction to smooth manifolds", or "Amann, Escher: ...
B.Hueber's user avatar
  • 2,836
5 votes
1 answer
943 views

Pulling back (affine or not) connections

Suppose that $f:M \to N$ is a smooth map of two manifolds and $E$ is a vector bundle over $N$. Given a connecton $\nabla$ on $E$ one can pull it back to $f^*N$ (which is a bundle over $M$) and obtain ...
truebaran's user avatar
  • 4,650
3 votes
1 answer
1k views

A description of line bundles on projective spaces, $\mathcal{O}_{\mathbb{P}^n}(m)$ defined using a character of $\mathbb{C}^*$.

I am trying to define line bundles on $\mathbb{P}^n$ as a triplet $E \xrightarrow{p} \mathbb{P}^n$. It follows the idea. Let $m \in \mathbb{Z}$ a fixed integer. Take $E$ the quotient of $\mathbb{C}^{n+...
wood's user avatar
  • 301
22 votes
1 answer
7k 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 ...
GFR's user avatar
  • 5,411
21 votes
2 answers
4k views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
hxhxhx88's user avatar
  • 5,277
20 votes
2 answers
12k views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Suppose $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ $c_2(V\otimes L)=...
HLC's user avatar
  • 1,119
17 votes
3 answers
8k 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 ...
User3568's user avatar
  • 644
17 votes
3 answers
4k views

When does a SES of vector bundles split?

Given a short exact sequence of smooth vector bundles, $$0\to A \to B \to C \to 0$$ on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on $...
Aaron's user avatar
  • 24.4k
14 votes
1 answer
947 views

Tangent bundles of exotic manifolds

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and $\phi^*...
Misha's user avatar
  • 401
14 votes
1 answer
2k views

Reduction of a structure group.

Let $X$ be a smooth manifold and $\pi:E\rightarrow X$ a vector bundle of rank $k$ on X. If one manages to redefine $E$ by using a cocycle $\{g_{\alpha,\beta}\}$ whose values are all contained in a ...
Pooya's user avatar
  • 185
9 votes
1 answer
2k views

Extension of Sections of Restricted Vector Bundles

Edit: Changing Question: There are two questions related questions: extending a smooth vector field extending a vector field defined on a closed submanifold I'm trying to answer a question which is a ...
Moya's user avatar
  • 5,288
8 votes
2 answers
2k views

Vector Bundle Locally Free Sheaf

Let $f: X \to Y$ a morphism of ringed spaces and $E$ a vector bundle over $Y$ for finite rank $r$. My questions are: Why $E$ can be interpreted as locally free sheaf of rank $r$, therefore for each $...
user267839's user avatar
  • 7,141
7 votes
1 answer
874 views

Existence of specific sections of vector bundles over a manifold

I am trying to do the following exercise from Hirsch, one could say that it's 3 exercises but they are all related so I believe it's best to treat them together: Let $\xi=(E,M,p)$ be an $n$-plane ...
Someone's user avatar
  • 4,757
6 votes
1 answer
1k views

Decompose vector fields on product manifolds

So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum $$ T_{(x,y)}(M \times N) = T_xM \oplus T_yN, $$ so that is obvious that the sum $X \oplus Y$ of smooth vector fields $...
deabo's user avatar
  • 557
5 votes
1 answer
1k views

Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$

On page $10$ of Hatcher's Vector Bundles and K Theory, he gives a proof that the Whitney sum of the trivial line bundle over $\mathbb{R}P^n$ and the tangent bundle is equal to the Whitney sum of ...
R Mary's user avatar
  • 1,852
4 votes
2 answers
820 views

The Affine Property of Connections on Vector Bundles

Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to \...
Rick's user avatar
  • 2,027
4 votes
1 answer
647 views

Why is the Legendre transform (of vector bundles) a smooth morphism $\mathbf FL:E\to E^*$?

The Legendre transform of convex functions $\mathbb R^n\to\mathbb R^n$ can be given a nice geometric interpretation as a way to characterise a function via the set of the tangent spaces to its graph. ...
glS's user avatar
  • 6,963
3 votes
1 answer
1k views

Direct proof of decomposition of real vector bundle of odd degree into the direct sum of a trivial bundle and another of even degree

The real splitting principle tells us that when taking a real, oriented vector bundle of odd dimension $\zeta$ over a manifold $M$ you can always write $\zeta$ as $\tilde{\zeta} \oplus \varepsilon^1$, ...
David Méndez's user avatar
1 vote
1 answer
483 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow. In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3)...
Thu An's user avatar
  • 31
1 vote
1 answer
240 views

Vector bundle and principal bundle

Why fiber of $(P\times V)/G\rightarrow P/G$ isomorphic to $V$ ? I think the fiber should be $V/G$, but it is not isomorphic to $V$ Picture below is from the 66 page of Jost's Riemannian Geometry and ...
Enhao Lan's user avatar
  • 5,931
23 votes
1 answer
3k views

If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?

Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
Jacob Bell's user avatar
20 votes
1 answer
8k views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
Patricio's user avatar
  • 577
15 votes
1 answer
2k views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
user avatar
14 votes
1 answer
895 views

Confusion about Positive Curvature in Holomorphic Bundles.

I'm trying to understand the principle that curvature decreases in holomorphic subbundles and increases in quotient bundles as shown in G-H (Griffiths Harris) page 78-79. Setup: Let $E\rightarrow M$...
Charles Ryder's user avatar
14 votes
1 answer
2k views

Why aren't complex vector bundles isomorphic to their duals?

For real vector bundles the argument goes like this: pick a metric on the bundle (i.e. a continuously varying metric on the fibers, which exists if the base space is paracompact), and map a vector $v$ ...
Emilio Ferrucci's user avatar
13 votes
1 answer
647 views

Lawson and Michelsohn's proof of the splitting principle for oriented vector bundles

I'm trying to understand the proof of the following result in Lawson & Michelsohn's Spin Geometry: Proposition 11.2. Let $E$ be an oriented real vector bundle of dimension $2n$ over a manifold $X$...
Jack Lee's user avatar
  • 47.4k
12 votes
1 answer
2k views

Does Hodge-star commute with metric connections?

Let $E $ be a smooth oriented vector bundle over a manifold $M$. Suppose $E$ is equipped with a metric $\eta$, and a compatible connection $\nabla$. Denote the dimension of $E$'s fibers by $d$. Let $\...
Asaf Shachar's user avatar
  • 25.3k
12 votes
1 answer
2k views

Are there vector bundles that are not locally trivial?

A $\Bbb K$ vector bundle over a space $B$ is a space $E$ and a continuous map $p:E\to B$ so that $p^{-1}(b)$ is a topological $\Bbb K$ vector space for any $b\in B$. One always includes local ...
s.harp's user avatar
  • 21.9k
11 votes
3 answers
1k views

Nonisomorphic vector bundles with diffeomorphic total spaces

What's an example of two non-isomorphic vector bundles $E,F$ over the same base such that the total spaces $Total(E), Total(F)$ are homeomorphic? Assume that rank of these bundles is the same as ...
Balarka Sen's user avatar
11 votes
1 answer
4k views

Isomorphism class of vector bundle over $\mathbb S^1$.

I'm currently self-studying through the book Differential forms in Algebraic Topology by Bott & Tu and got stuck on an exercise 6.10, which asks to compute $\textrm{Vect}_k(\mathbb S^1),$ the ...
JCM's user avatar
  • 575
11 votes
1 answer
5k views

Determinant of a tensor product of two vector bundles

Let $X$ be a smooth variety over a field, $V_1$ and $V_2$ are two vector bundles over $X$ of ranks $r_1$ and $r_2$ respectively. Determinant of a vector bundle is the top exterior power of the vector ...
Alex's user avatar
  • 6,284
10 votes
1 answer
2k 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 ...
user136885's user avatar
10 votes
2 answers
7k views

Canonical sheaf projective bundle

Given a smooth variety $Y$, and a vector bundle $E$ of rank $r+1$ defined on it, call $X$ the variety associated to the projective bundle given by $E$. Call $\pi: X \rightarrow Y$ the natural map. ...
Stefano's user avatar
  • 4,364
9 votes
1 answer
1k views

Isomorphism between $\operatorname{Hom}(E,F)$ and $E^*\otimes F$, $E$ and $F$ vector bundles.

I need some help in finding the solution to the following problem: Let $E$, $F$ be vector bundles (of finite rank, if needed) over a manifold $M$. Consider the set $\operatorname{Hom}(E,F)$ of all ...
Marra's user avatar
  • 4,869
9 votes
1 answer
889 views

Functorial properties of the degree in vector bundles

Let $X$ be a Riemann surface$^\dagger$ and consider a complex vector bundle $E$ over it. I know that the definition of the degree of the bundle $E$ is given by $$\text{deg} E = \frac{i}{2\pi} \int_X \...
topolosaurus's user avatar
  • 1,943
9 votes
4 answers
614 views

Why do we need the local triviality condition when working with vector bundles?

I'm currently reading through Adams' paper on the image of the J homomorphism, and wanted to brush up on vector bundles and K-theory before tackling this paper. The definition of (real) vector bundle ...
Caleb Miller's user avatar

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