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 $X$.
268
questions
30
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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 ...
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votes
5
answers
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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 ...
- 127k
2
votes
1
answer
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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}...
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votes
2
answers
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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 ...
- 153k
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votes
4
answers
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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?
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votes
3
answers
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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)$ (...
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votes
2
answers
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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, ...
- 5,185
12
votes
1
answer
878
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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$...
- 13.7k
28
votes
1
answer
6k
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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) \...
- 1,897
19
votes
4
answers
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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 ...
- 32.3k
12
votes
1
answer
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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 ...
- 17.6k
9
votes
1
answer
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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 ...
- 3,254
21
votes
2
answers
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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$$
So is ...
- 5,129
20
votes
1
answer
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Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?
Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts ...
- 14.2k
20
votes
1
answer
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How a principal bundle and the associated vector bundle determine each other
It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...
- 5,231
19
votes
2
answers
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Quick question: Chern classes of Sym, Wedge, Hom, and Tensor
Suppose $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$:
$c_1(V\otimes L)=c_1(V)+rc_1(L)$
$c_2(V\otimes L)=...
- 1,089
15
votes
3
answers
4k
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When does a SES of vector bundles split?
Given a short exact sequence of smooth vector bundles,
$$0\to A \to B \to C \to 0$$
on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on $...
- 23.4k
13
votes
1
answer
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Reduction of a structure group.
Let $X$ be a smooth manifold and $\pi:E\rightarrow X$ a vector bundle of rank $k$ on X. If one manages to redefine $E$ by using a cocycle $\{g_{\alpha,\beta}\}$ whose values are all contained in a ...
- 175
8
votes
2
answers
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Why is the tangent bundle defined using a disjoint union?
In textbooks about differential geometry, one finds often the disjoint union in the definition of the tangent bundle (e.g. in "Lee: Introduction to smooth manifolds", or "Amann, Escher: ...
- 1,992
7
votes
1
answer
490
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Existence of specific sections of vector bundles over a manifold
I am trying to do the following exercise from Hirsch, one could say that it's 3 exercises but they are all related so I believe it's best to treat them together:
Let $\xi=(E,M,p)$ be an $n$-plane ...
- 4,456
6
votes
1
answer
302
views
Defining Connection in Hom Vector Bundle without Coordinates, using Dieudonne's Definition
In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given:
A linear connection $C$, in a vector bundle $(E,\pi, M)$ ...
- 43.9k
6
votes
1
answer
886
views
Decompose vector fields on product manifolds
So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum
$$
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 $...
- 475
5
votes
2
answers
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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 $...
5
votes
1
answer
1k
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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 ...
- 1,792
4
votes
2
answers
714
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 \...
- 1,977
3
votes
1
answer
927
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$, ...
- 163
1
vote
1
answer
209
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 ...
- 5,507
1
vote
1
answer
401
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)...
- 31
16
votes
1
answer
6k
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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 ...
- 507
14
votes
1
answer
760
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$...
- 1,387
14
votes
1
answer
2k
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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 ...
user641
13
votes
1
answer
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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$ ...
- 2,484
12
votes
1
answer
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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 ...
- 20.9k
12
votes
1
answer
822
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^*...
- 371
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votes
1
answer
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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 $\...
- 24.4k
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votes
3
answers
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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 ...
- 13.6k
10
votes
1
answer
4k
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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 ...
- 5,960
9
votes
1
answer
399
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 \...
- 1,753
9
votes
1
answer
2k
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Pull-back of sections of vector bundles
I'm sure this is a silly question but I'm stuck at the concept of pulling back sections of a vector bundle. Let $\pi:E\to X$ be a vector bundle on a variety $X$ and $f:Y\to X$ a morphism. We have a ...
- 13.7k
9
votes
1
answer
450
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 $X$...
- 43.8k
9
votes
1
answer
1k
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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 ...
- 91
8
votes
1
answer
956
views
Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G
If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
- 2,589
8
votes
2
answers
2k
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Tangent bundle : is a manifold
I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
- 191
8
votes
1
answer
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If a line bundle admits a non-vanishing section then it is trivial
Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$.
I have to prove that the line bundle above is trivial.
Idea: ...
- 2,727
8
votes
3
answers
5k
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Understanding the trivialisation of a normal bundle
I've been looking for a definition of "trivialisation of normal bundle".
I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
- 44.9k
8
votes
2
answers
2k
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Triviality/non-triviality of line/circle bundle over $S^3$
I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
- 81
8
votes
1
answer
895
views
Rank $n$ vector bundle with $n$ pointwise linearly independent sections is trivial
I have an $n$-dimensional vector bundle $E \to X$ and sections $s_1, \dots, s_n : X \to E$ such that for any $x \in X$, the elements $s_1(x), \dots, s_n(x)$ are linearly independent in the vector ...
- 383
8
votes
1
answer
916
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 ...
- 4,704
8
votes
1
answer
3k
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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 ...
- 523
7
votes
2
answers
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"Basis extension theorem" for local smooth vector fields
Let $\pi: E \to M$ be a smooth vector bundle of rank $n$, and suppose $s_1, \ldots, s_m$ are independent smooth local sections over an open subset $U \subset M$.
Can I prove the "basis extension ...
- 561