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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exterior product resolutions of locally free sheaves
Consider an exact sequence $0\rightarrow\mathcal{E}\rightarrow\mathcal{F}\rightarrow\mathcal{G}\rightarrow0$ of locally free sheaves on a projective manifold $X$, then one has the following two "...
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Geometric meaning of the grading of a module
I know that a $\mathbf Z$-grading of a $k$-algebra $S$ is the same thing as a group-action of $(\mathbf G_m)_k$ on $Spec(S)$. I was wondering if there is a similar nice geometric description of a $\...
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Understanding the tensor bundle
On page 13 in Hatchers book the tensor product operation on vector bundles is introduced. Let $p_1:E_1 \to B, p_2:E_2 \to B$ be two vector bundles. Then $E_1 \otimes E_2$ is defined to be $E_1 \otimes ...
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On the positivity of the second Segre class of ample vector bundles
Let $E$ be an ample rank $r\geq2$ vector bundle over a smooth projective surface $X$ defined on an algebraically closed field $\mathbb{K}$ of characteristic $0$.
In Kleiman S. L. - Ample Vector ...
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A connected compact manifold has a unique orientable line bundle on it
How to show that if $M$ is a connected compact manifold there is upto isomorphism a unique orientable line bundle on it and that the line bundle is trivial iff $M$ is orientable?
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A (maybe) trivial question on trivial vector bundles: alternative definition of trivial vector bundle
I am studying Loring W. Tu's Differential geometry, Connections, Curvature and Characteristic Classes and I am having a doubt (the same doubt I had when studying the same topic in the author's An ...
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Double covariant derivative in coordinates: Why does this work?
Lets take a vector field $X$ on some Riemannian manifold $(\mathcal{M},g)$ with Levi-Civita connection $\nabla$. Then, the components of its covariant derivative in coordinates are
$$\nabla_{\alpha}X^{...
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Are Functions Smooth Sections?
In differential geometry we often identify $\Omega^0(M)$ with the smooth functions on
a smooth manifold $M$. But for every $i>0$ we know that:
$$\Omega^i(M)=\Gamma(\Lambda^i(T^*M))$$
I imagine with ...
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More details in the proof of the equivalence of categories between vector bundles over a smooth manifold and locally free sheaves
$\def\VB{\mathsf{VB}}
\def\sO{\mathcal{O}}
\def\Mod{\mathsf{Mod}}
\def\LFMod{\mathsf{LFMod}}$For a ringed space $(X,\sO_X)$ and a ring $R$, denote $\Mod(\sO_X)$ and $\Mod(R)$ to the categories of $\...
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Sasaki metric on vector bundle and horizontal distribution
We can equip the tangent bundle $\pi_{TM} : TM \to M$ with the Sasaki metric
\begin{equation*}
ds^2 = g_{ij}dx^idx^j + g_{ij}Dv^iDv^j,
\end{equation*}
where $Dv^i = \Gamma^i_{jk}v^jdx^k$ denotes the ...
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Vector space structure on fibre of line bundle
Let $I=[0,1]$ and $E=I \times \mathbb{R}/\sim,$ where $\sim$ identifies $(0,t) \sim (1,-t)$. The projection $I \times \mathbb{R} \to I$ induces $p:E \to S^1$. This is supposed to be a $1$-dimensional ...
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Vector bundles on $S^2$ up to isomorphism [duplicate]
I've recently become acquainted with the notion of a vector bundle on a topological space. I'm interested in what the vector bundles are over $S^2$. Certainly, it admits the tangent bundle. ...
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Sections of pullback bundle are generated by pullback of sections
Let $M,N$ be smooth manifolds, $\pi:E\to M$ a smooth rank $r$ vector bundle, $f:N\to M$ a smooth map. We define
$$
f^*E=\{(p,v)\in N\times E: f(p)=\pi(v)\},
$$
which has a natural structure of a ...
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Are there such things as "function bundles", analogous to vector bundles?
I am interested in whether or not the typical vector bundle construction can be extended to cases where each fiber is an infinite dimensional vector space, possibly with other structure associated ...
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Confusion in notation of representation of Bastiani derivative
In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions
Definition II.2. Let $U$ be an open subset of a ...
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Trying to construct a smooth inverse for a smooth map I know to be a diffeomorphism.
Let $M$ be a smooth manifold, and $\text{Fr}_{GL}(M)$ be the frame bundle. Let $\rho$
be the standard representation of $GL_{n}(\mathbb{R})$ on $\mathbb{R}^n$. I am trying to show that the associated ...
- 1,137
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Sheaves and bundle morphisms
I'm currently trying to reunderstand differential geometry using shaves. Let $f:(E_1,M,\pi_1)\rightarrow (E_2,M,\pi_2)$ be a $\mathcal{C}^{k}$ vector bundle morphism, where $M, E_1$ and $E_2$ are a $\...
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Non-degeneracy of the Poisson structure.
I am reading a book on Poisson manifold. There I found the notion of non-degenerate Poisson structure.
Definition $:$ Let $M$ be a Poisson manifold with Poisson bivector field $\pi.$ Then $\pi$ ...
- 2,647
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Structure on moduli space of topological/smooth vector bundles
If $M$ is paracompact (let us assume smooth manifold of dimension $d$), then one has that the set of isomorphism classes of vector bundles of rank $n$ over $M$ is isomorphic to the set of homotopy ...
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Space of smooth section of a vector bundle as Hilbert spaces
Currently, I am studying the space of sections $\Gamma(E)$ of a vector bundle $(E,M,\pi)$. If $h$ is a riemannian bundle metric over $E$, and $\mu$ is a volume form in $M$, we can define the following ...
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Why does every vector bundle admit a map into the universal bundle?
The following question regards a proof about Lemma 5.3 in Milnor-Stasheff. The proposition is to prove that every rank $n$ vector bundle $\pi:E\to B$ with compact (more generally, paracompact) base ...
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Curvature of Connection [exercise proofcheck]
I tried to solve the following question, but I get the feeling I am doing a few things wrong, but cannot really find out what.
So the exercise is:
Let $\alpha, \beta \in \Omega^{1}(M)$ for a manifold $...
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Reference for Schwartz kernel theorem on vector bundles
In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
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Correctness of proof of Vector Bundle Construction Theorem
I want to check if this proof is correct. I have seen this post that constructs an equivalence relation on the set $\coprod\limits_{\alpha \in A}U_\alpha \times \mathbb{R}^k$, however my approach ...
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Confusion about basis of Endomorphism bundle
Excuse me if this is a silly, question, but I seem to miss something obvious.
From my knowledge, given a finite-dimension vector bundle $E$ over $M$, we can always trivialize it over opens $U_{\alpha}$...
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Realization of the $2$-form as the curvature of some bundle.
Suppose $M$ be a manifold, and $\omega$ be any smooth $2$-form. My question is about the existence of the Lie group $G$ which satisfies that there is always a principal $G$-bundles $P$(or its ...
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Bundles with large rank
Suppose $E\to M$ is a vector bundle of rank $k$ bundle over an n-dimensional manifold. If $k>n$, is there a splitting $E\cong F\oplus G$ where the rank of $F$ is $n$ and the rank of $G$ is $k-n$? ...
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Are measurable bundles fiber bundles?
I bumped into a definition of measurable bundle from page 101 of an introduction to smooth ergodic theory by Barreira and Pesin.
Let $E$, $X$ be measurable spaces and $\pi:E\to X$ be a measurable ...
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Difference between $J_{0}^{1}E$ and $\hat{J}_{0}^{1}E$ for some vector bundle $E$ over $M$
Apologies if this is an obvious question (I'm self learning this with applications to physics)
Suppose we have some vector bundle over a Riemannian manifold M defined by \pi:E\rightarrow M. If we look ...
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Real Line Bundle Corresponding to a Double Cover
I am studying spin structures on $SO(n)$-bundles using some lecture notes. Right after defining the twist of a spin structure $(P,\psi,\rho)$ on $Q\xrightarrow{} X$ by a double cover $\pi:R\...
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Classification of vector bundles
Let $G_n$ be the infinite Grassmanian and $E_n$ the canonical $n$-vector bundle over $G_n$. For a paracompact $X$, the map $[X,G_n] \to \operatorname{Vect}^n(X)$ sending the homotopy class $[f]$ to ...
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Complex conjugation of bundle-valued forms
Consider a holomorphic vector bundle $V$ over an $n$_dim complex manifold $M$. I am interested in the notion of an inner product between complex bundle-valued forms $\phi^a, \psi^b \in \Omega^{(p,q)}(...
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Euler number of symmetric cube of the tautological bundle
Let $E$ -is a tautological two-dimensional bundle (rank $n=2$) over complex Grassmannian $\operatorname{Gr}(2, 4)$ ($2$-dimensional planes in $C^4$). I'm trying to compute the Euler number $\oint_{\...
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analytic discription of $\mathrm{Ext}^1$
We know that for any pair of holomorphic vector bundles $\mathcal{E}_1$ and $\mathcal{E}_2$ over complex surface $X$(i.e. complex dimension 2), $\mathrm{Ext}^1(\mathcal{E}_1, \mathcal{E}_2)$ can be ...
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How the k-th Chern class looks like?
I have a question about the definition of the Chern class. I have studied the subject via the introduction of invariant polynomials ("Loring W.Tu Differential Geometry").
Let $P(X)$ a ...
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transition functions on vector bundles
If we consider a vectorbundle $\pi:V\rightarrow M$ on a smooth manifold and two open subsets $U_1,U_2$ along with isomorphisms $h^{U_i}:\pi^{-1}(U_i)\rightarrow U_i\times \mathbb{R}^n$ s.t. $pr_1 \...
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Is the space of section of a vector bundle a diffeological space?
Let $(E,M,V,\pi)$ be vector bundle over $M$ with fiber $V$ and projection $\pi$.
I am tryng to prove that the space of sections $\Gamma(E)$ of this vector bundle is a diffeological space. I denote the ...
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Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on $T\mathbb{C}P^1$
My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves
$$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$
I want to ...
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Almost vector bundle with no local trivialization
A (smooth real) vector bundle over an $m$-dimensional smooth manifold $M$ is a triple $(E,M,\pi)$, such that
$E$ and $M$ are smooth manifolds
$\pi$:$E\rightarrow M$ is a submersion.
The fibres $E_p:...
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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+...
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Problem in undersanding notation of distributional section
In this post Observables By Urs Schreiber he denotes the space of distributional sections
let $\Gamma_{\Sigma}^{\prime}\left(E^*\right):=\left(\Gamma_{\Sigma, c p}(E)\right)^*$ in the definition ...
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Definition of vector bundle
Let $E$, $B$, $F$ be topological spaces, and $p:E\to B$ a continuous surjection. These data define a fiber bundle with fiber $F$ if, for every $b\in B$, exists an open set $b\in U\subset B$ and a ...
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Extending a section of vector bundle to union of open sets
I'm having some trouble with one part of Bott–Tu Theorem 6.8, which proves that homotopic maps induce isomorphic vector bundles under pullback.
The setup is that $\pi:Y\times I\to Y$ is the projection,...
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Pushforward of a line bundle on $\mathbb{P}^1$ to a point
In his notes on K-theory, A. Okounkov states the following exercise:
The group $\operatorname{GL}(2)$ acts naturally on $\mathbb{P}^1$ and on line bundles $\mathcal{O}(k)$ over it. Push forward these ...
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Regular homotopy and the weak Whitney embedding theorem
Two maps $f$ and $g\in$ $Imm(M,N)$ are regular homotopic if they are homotopic via immersions $h_t:M \to N$ and the derivatives $Th_t:TM
\to TN$ of $h_t$ define a homotopy of bundle monomorphisms $$...
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Are trivializations always linear?
In Tu's Introduction to Manifolds he proves that if $s$ and $t$ are smooth sections of a smooth vector bundle $\pi: E \rightarrow M$ and $f$ a smooth real-valued function on $M$, then $s + t$ and $fs$ ...
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Lie group representation associated to a group object in the category of smooth vector bundles
It is mentioned in the introduction of the paper VB-GROUPOIDS AND REPRESENTATION THEORY OF LIE GROUPOIDS by LFONSO GRACIA-SAZ AND RAJAN AMIT MEHTA https://arxiv.org/pdf/1007.3658.pdf that there is a ...
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Chern classes of the tautological vector bundle of a product of a project space
Consider the projective variety $\mathbb{P}(\mathbb{C}^n)^r = \mathbb{P}(\mathbb{C}^n) \; \times \; ... \times \; \mathbb{P}(\mathbb{C}^n) \; (r \text{ times})$, where $r \leq n$. We have a surjection
...
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Product in the category of smooth vector bundles
Is the product in the category of smooth vector bundles just the direct product of smooth vector bundles?
More precisely, if $\pi:E \rightarrow B$ and $\pi': E' \rightarrow B'$ are two smooth vector ...
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Importance of studying the curvature of a covariant derivative on a vector bundle
Let $\mathcal{E}$ be a vector bundle over $M$. Let $\nabla : \Gamma (M,\mathcal{E}) \rightarrow \Gamma(M,T^*M \otimes \mathcal{E})$ be a covariant derivative on M. I read that the choice of a ...
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