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 $X$.
2,792
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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 ...
- 303
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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 ...
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66
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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 $q\in M$ ...
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63
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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 ...
- 2,193
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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 ...
- 2,460
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1
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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 $C$, ...
- 361
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62
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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 ...
- 73
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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 $p:E\to M$ ...
- 500
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33
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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 ...
- 326
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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 ...
- 41
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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 ...
- 26.1k
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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 ...
- 3
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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 $\{...
- 171
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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 ...
- 111
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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....
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2
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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 ...
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$F \subset E$ 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 $...
- 121
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1
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33
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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 ...
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69
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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 ...
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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) $ ...
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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 ...
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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 $S$ ...
- 456
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48
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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 ...
0
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15
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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. ...
- 924
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2
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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 ...
- 1,329
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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!!
- 352
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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 ...
- 1,791
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1
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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 \...
2
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50
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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 ...
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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 ...
- 4,805
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1
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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 ...
- 371
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1
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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 ...
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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 $\mathrm{GL}_n(\mathcal O)\backslash \mathrm{GL}_n(\Bbb A)/\mathrm{GL}_n(K)$, ...
- 19.9k
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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 ...
- 1,369
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28
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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) ...
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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 ...
0
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1
answer
43
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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 ...
- 327
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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)= \{(...
- 128
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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,...
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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 ...
- 969
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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 ...
- 924
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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\...
- 958
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10
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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 $...
- 91
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1
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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 ...
- 2,990
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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 $E$ ...
- 7
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1
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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 ...
0
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1
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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 ...
- 7
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1
answer
140
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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 ...
- 163
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1
answer
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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 ...
- 1,031
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1
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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 ...
- 4,756