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 $X$.

Filter by
Sorted by
Tagged with
2 votes
0 answers
20 views

Is a monomorphism of vector bundles an embedding?

Let $(p,E,B)$ and $(p',E',B)$ be two $C^r$ vector bundles over the same base space $B$. (When $r>0$ all the spaces are $C^r$ manifolds and all the maps are $C^r$ smooth. When $r=0$ they are just ...
Skyskie's user avatar
  • 303
0 votes
0 answers
21 views

Elementary proof of Bott periodicity

The elementary proof of Bott periodicity was found by Atiyah and Bott, and published in the paper "On the periodicity theorem for complex vector bundles". In Hatcher's book p.41-51, this ...
Dasheng Wang's user avatar
0 votes
0 answers
66 views

Canonical chart for parallelizable manifold

Let $M$ be an $n$ dimensional connected parallelizable manifold and $\{X_1,\dots,X_n\}$ a basis for the tangent bundle. Consider the following Claim. There existis $p\in M$ such that for all $q\in M$ ...
Gabriel Golfetti's user avatar
1 vote
0 answers
63 views

Apparent contradiction to Atiyah-Krull-Schmidt

Let $V$ be a $k$-vector space equipped with an anisotropic bilinear form $B\colon V\otimes V\to k$, e.g. with $k=\mathbb{R}$ we can let $B$ be any inner product on $V$. Let $L$ be the tautological ...
BHT's user avatar
  • 2,193
5 votes
1 answer
200 views

Algebraic vector bundles which are analytically but not algebraically isomorphic

I am looking for an example of two algebraic vector bundles on an algebraic complex manifold / smooth complex algebraic variety which are analytically isomorphic, but not algebraically isomorphic. By ...
Earthliŋ's user avatar
  • 2,460
4 votes
1 answer
72 views

Propositions 10.12 and 10.15 from Lee's book "Introduction to Smooth Manifolds"

Let $\pi:E\to M$ be a (smooth) vector bundle. 10.12. Let $C$ be a closed subset of $M$ and $s:C\to E$ a (smooth) section of $\pi_C:E|_C\to C$. For each open subset $U \subseteq M$ containing $C$, ...
user1234567890's user avatar
0 votes
0 answers
62 views

Uniqueness of Levi-Civita connection in terms of horizontal distributions

Lately, I have been wondering about how to make the reasoning of uniqueness of the Levi-Civita connection $\nabla$ on a tangent bundle $\pi:TM\to M$ with a bundle metric $g$, via f.e. the Koszul ...
whatever's user avatar
1 vote
1 answer
75 views

A question related to frame bundle of a vector bundle

I am currently reading up on principal fiber bundles from a set of lecture notes on the subject, and I am trying to make sense of frame bundle of a vector bundle. Consider a vector bundle $p:E\to M$ ...
neophyte's user avatar
  • 500
1 vote
0 answers
33 views

The existence of global continuous frame implies the existence of global smooth frame

I am trying to show that if there exists a global continuous frame for a smooth vector bundle over a smooth manifold $M$, then we can find a global smooth frame. First, I have proven the simple ...
Quzs's user avatar
  • 326
2 votes
0 answers
90 views

Equivalance of Vector Bundle Definitions

I try to learn about manifolds and I am working with the two textbooks "An Introduction to Manifolds" by Tu and Lee's book "Introduction to Smooth Manifolds". After comparing the ...
RobRTex's user avatar
  • 41
2 votes
1 answer
72 views

Can we use the pullback connection to induce a connection on the tangent bundle of a submanifold?

Let $M$ be a smooth manifold. Let also $\nabla$ be an affine connection on $TM$. If $\Sigma\subset M$ is an embedded sub manifold and $\iota:\Sigma\to M$ is the inclusion we can construct the pullback ...
Gold's user avatar
  • 26.1k
0 votes
0 answers
28 views

Transition functions for the cotangent bundle

In R.W.R Darling, on differential geometry, an exercise is to construct the cotangent bundle $T^{*}M$ from transition functions $g_{\gamma\alpha}(p)$. A quick analysis suggest using equivalence ...
Bin's user avatar
  • 3
0 votes
0 answers
37 views

Are local smooth sections sufficient to conclude we have a vector bundle?

Suppose we have a submersion $\pi: E \to M$ such that for each $x \in M$ the fibre $E_x$ is a vector space. Moreover, suppose that for every $x \in M$ we have a collection of smooth local section $\{...
Yadeses's user avatar
  • 171
0 votes
0 answers
29 views

Holonomy vs structure group of tangent bundle

Given a Riemannian manifold $(M,g)$, is the holonomy group of $g$ the structure group of the oriented tangent bundle on $M$? If not, is there a relation between the two? Their definitions seem quite ...
user984949's user avatar
0 votes
0 answers
51 views

Dual Representation and Dual Vector Bundle

The lecture notes I used in my differential geometry class defined attaching to $P$ the fiber $V$ (w.r.t $P$) as follows: We define the vector bundle obtained by attaching to $P$ the fiber $V$ (w.r.t....
Z.Y.H's user avatar
  • 121
2 votes
2 answers
111 views

Rank of vector bundle homomorphism is almost constant

Let $M$ be a smooth manifold, $\pi : E \to M$ and $\pi' : E' \to M$ smooth vector bundles and $f : E \to E'$ a smooth bundle map, i.e. $\pi' \circ f = \pi$ and the restriction of $f$ to each fibre is ...
FJerotte's user avatar
1 vote
0 answers
13 views

$F \subset E$ as a sub-vector bundle of E, Show that it is a $GL_p$-reduction of $Fr(E)$

Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $...
Z.Y.H's user avatar
  • 121
0 votes
1 answer
33 views

Confusion about vector bundle extension lemma from Atiyah's K-Theory

Lemma 1.4.2 from Atiyah's K-theory book (p16-17): Let $Y$ be a closed subspace of a compact Hausdorff space $X$, and let $E,F$ be two vector bundles over $X$. If $f:E\vert_Y \to F \vert_Y$ is an ...
jasone's user avatar
  • 101
4 votes
1 answer
69 views

Complex vector bundle valued forms

Let $X$ be a smooth manifold and $E \to X$ a complex vector bundle. A connection on $E$ is a $\mathbb{C}$-linear map $\nabla \colon \Gamma(X,E) \to \Omega^1(X,E)$ where $\Gamma(X,E)$ denotes the space ...
KuSi's user avatar
  • 151
1 vote
1 answer
80 views

Question on Segal's definition of $K$-theory

Segal, in the papers "Fredholm complexes" and "Equivariant K-theory", gives the following equivalent definitions of $K$-theory. For $X$ a compact top. space, let $\mathcal{L}(X) $ ...
Overflowian's user avatar
  • 5,540
0 votes
0 answers
16 views

More intrinsic definition of secondary vector bundle structure

The Wikipedia definition of Secondary vector bundle structure is based on local coordinates. Maybe a more intrinsic definition can be given. The idea is the same as what Wikipedia does. $(E,p,M)$ is a ...
jw_'s user avatar
  • 489
0 votes
0 answers
50 views

Normal bundles and tubular neighborhoods

Let $M$ be smooth manifold and $S^s \hookrightarrow M^m$ a smooth embedding. Denote the normal bundle of $S$ in $M$ via $N_{S/M}$. A tubular neighbourhood (of $S$ in $M$) is a disc bundle over $S$ ...
ferhenk's user avatar
  • 456
2 votes
0 answers
48 views

Ereshamann Connection on Sphere's Frame Bundle

In order to better understand connections on principal bundles, I would like to compute the Ereshmann distribution (horizontal distribution) and the connection 1-form associated on the frame bundle of ...
Alessandro Tamai's user avatar
0 votes
0 answers
15 views

Sufficient conditions for a vector bundle to be an exterior bundle of some vector bundle

Given a smooth rank $2^n$ real vector bundle $\pi:E\to M$ over a smooth manifold $M$. I want to determine sufficient conditions for $E$ to be isomorphic to an exterior bundle of some vector bundle. ...
Uncool's user avatar
  • 924
2 votes
2 answers
56 views

Understanding proof that $E$ and $E^\ast$ are isomorphic rank $1$ bundles.

I would like to prove the following: Proposition. Let $E$ be any real line bundle over $M$. Then $E$ and $E^\ast$ are isomorphic line bundles. I have sketched what I believe works, but am having ...
Rough_Manifolds's user avatar
0 votes
0 answers
58 views

Nontrivial microbundles

I'm looking for easy examples of nontrivial microbundles (meaning: microbundles which are not stably isomorphic to the trivial microbundle). Some examples on $S^1$ or even $S^n$ would be really nice!!
groupoid's user avatar
  • 352
0 votes
0 answers
13 views

Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:E\rightarrow X$ be a holomorphic vector bundle of rank $n$. Given an orthogonal form on $E$, i.e. a symmetric non-degenerate bilinear ...
Hajime_Saito's user avatar
  • 1,791
1 vote
1 answer
65 views

Stability on short exact sequences of vector bundles

I have the following problem: consider a complex algebraic curve $C$ and $L,E,M$ slope semi stable vector bundles on $C$ that fits in a short exact sequence as following: $$ 0 \rightarrow L \...
Erick David Luna Núñez's user avatar
2 votes
0 answers
50 views

Why fiber bundles are locally products

I'm a first year graduate student in math and I'm currently studying fiber bundles. The definition is clear and I understand how it generalize concepts as (co)tangent bundles or vector bundles. What ...
Gesture Glove's user avatar
1 vote
0 answers
70 views

Notation in the local form of a covariant derivative

Let $\pi: P \rightarrow M$ be a principal $G$-bundle with connection $A$ and $E = P \times_\rho V$ an associated vector bundle to $P$ where $V$ is a vector space and $\rho: G \rightarrow GL(V)$ is a ...
CBBAM's user avatar
  • 4,805
0 votes
1 answer
69 views

Is there an easy direction for the Higgs correspondence?

There is a deep famous correspondence between analytic and algebraic properties. For a complex curve $X$, representations $Hom(\pi_1(X), U(n))$ correspond to degree $0$ semistable bundles. This is a ...
user135743's user avatar
1 vote
1 answer
81 views

Invariant connection on principal bundle

Suppose I have a principal bundle, and some group G acting on the principal bundle. Is it always possible to find a G-invariant connection on the principle bundle? If G is compact, then I can imagine ...
jaws93's user avatar
  • 11
0 votes
1 answer
128 views

Vector Bundles on a Curve as an Adelic Double Quotient

I'm looking for an explanation or a reference on why rank $n$ vector bundles on a curve over a field can be described as $\mathrm{GL}_n(\mathcal O)\backslash \mathrm{GL}_n(\Bbb A)/\mathrm{GL}_n(K)$, ...
Lukas Heger's user avatar
  • 19.9k
2 votes
0 answers
87 views

Determinant bundle of $(p,q)$ forms

My actual question is much simpler but I would be interested to what happens in a generic case as well. Here's three cases which I want to know in general: $X$ Kähler 3-fold, what are the determinant ...
Partha's user avatar
  • 1,369
0 votes
0 answers
28 views

Metric that makes distribution orthogonal to each other

Let $M$ be a manifold and $Q\subseteq M$ a submanifold. Suppose that there is an integrable distribution (sub-bundle) $E$ of $TM|_Q$ such that $TM|_Q=TQ\bigoplus E$. Can we guarantee a (Riemannian) ...
Fernando Nazario's user avatar
0 votes
0 answers
42 views

Equivalences of different perspectives of a connection on a principal bundle

A connection $A$ on $P$ can be viewed in many ways: A $G$-invariant 'horizontal' subbundle $H_A \subset T P$ transversal to the vertical tangent space $V T P=\operatorname{ker}(d \pi)$; a system of ...
Paul Immanuel's user avatar
0 votes
1 answer
43 views

Sections of submersions with values in a distribution

I am entertaining the following scenario: It is well known that given any surjective submersion $p\colon M\rightarrow N$ between smooth manifolds (without boundaries), one can always find a local ...
Žan Grad's user avatar
  • 327
0 votes
0 answers
25 views

What is the smooth structure on the Frame Bundle?

Let $M$ be a manifold of dimension $n$ and let $\pi :E \rightarrow M$ be a (real) vector bundle of rank $r$, we want to associate a $GL_r-$ principal bundle to it in the following way: Let $Fr(M)= \{(...
Armando Patrizio's user avatar
1 vote
1 answer
97 views

Existence of complex line subbundle of complex bundles

Suppose $M$ be a real smooth manifold, and let $E$ be a rank $r>1$-complex vector bundle over $M$. I would like to know that if there is a complex line subbundle of $E$. By usual obstruction theory,...
ChoMedit's user avatar
  • 744
1 vote
0 answers
43 views

How does a smooth functor guarantee the existence of another smooth vector bundle from a given one?

I am trying to understand the following definition of smooth functors on the category of finite dimensional vector spaces (in the context of smooth vector bundle theory): Let $\mathsf{Vec}$ be the ...
math-physicist's user avatar
0 votes
2 answers
67 views

A vector bundle whose fibers are isomorphic to an exterior algebra of a vector space is an exterior bundle.

Given a smooth vector bundle $\pi: E\to M$ it is a standard result that the exterior vector bundle $\pi':\wedge E\to M$ where $\wedge E= \sqcup_{p}\wedge E_p$ is also a smooth vector bundle. My ...
Uncool's user avatar
  • 924
0 votes
0 answers
46 views

Reference on identifying the differential of a map with a $(1,1)$ tensor field

I've seen several times on the Internet the pretty cool expression that \begin{equation}\label{1}\mathrm df=\sum_{i,j}{\partial f^i\over \partial x^j}{\partial \over \partial y^i}\otimes\mathrm dx^j\...
painday's user avatar
  • 958
0 votes
0 answers
10 views

Index and null of a hermitian metric

I'm following Kobayashi's Differetianl Geometry of Complex Vector Bundles. In section IV.3 he shows a result regarding a hermitian metric $h$ on a holomorphic bundle $E$, and says something about $...
ken ruiz's user avatar
2 votes
1 answer
57 views

Structure sheaf of the projectivization of a bundle

Let $0\to V'\to V\to M \to 0$ be an exact sequence with $V$ a vector bundle over a scheme $B$, $M \in \operatorname{Pic}(B) $. Let $H:=\mathbb{P}(V')$ be the Cartier divisor, let $p: \mathbb{P}(V) \to ...
Conjecture's user avatar
  • 2,990
1 vote
0 answers
63 views

Vector Bundles Categorically

Definition I understand the first two points of this definition. However I need assistance in understanding further the sentance: It is required that the topology of $E_x$ as a subspace of $E$ ...
S. Marco's user avatar
2 votes
1 answer
154 views

topology on smooth vector bundle

Let $\pi: E\to M$ be a smooth vector bundle. Here $E$ and $M$ are smooth manifolds. Wikipedia (and many others) says that the maps $\varphi: U\times \mathbb{R}^k\to \pi^{-1}(U)$ are the local ...
Laurent Claessens's user avatar
0 votes
1 answer
69 views

Category of Vector Bundles

I am doing Category Theory and part of my project is to understand the category of vector bundles. Is there any reference(s) for vector bundles from a categorical perspective? I also aim to understand ...
S. Marco's user avatar
1 vote
1 answer
140 views

Klyachko's classification of toric vector bundles - an ongoing discussion

I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra) I am ...
sagirot's user avatar
  • 163
2 votes
1 answer
76 views

Reference for the following cancellation theorems

I found these two cancellation theorems in the K-book, in section 1.4, page 39: Real Cancellation Theorem: Suppose $X$ is a $d$-dimensional CW complex and $\eta:E\to X$ is an $n$-dimensional real ...
SummerAtlas's user avatar
  • 1,031
0 votes
1 answer
31 views

A couple of questions regarding the Vector Bundle Chart Lemma

I have a couple of questions regarding the proof of the Vector Bundle Chart Lemma in Lee's Introduction to Smooth Manifolds: Why is $\mathbb{H}^n\times\mathbb{R}^k\cong\mathbb{R}^{n+k}$, and what is ...
Sam's user avatar
  • 4,756

1
2 3 4 5
56