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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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Integration along fibers is independent of the lift

According with Wikipedia https://en.wikipedia.org/wiki/Integration_along_fibers Let $\pi: E \to B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$ is a $k$-form on $E$, ...
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Reduction of structure group

First, this question has been asked before, but no one really answered it. Let $p:E\to B$ be a rank $n$, real vector bundle with transition functions $\{g_{UV} : U\cap V \to Gl_n(\mathbb{R})\}$. I ...
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The dual of $C_0(X,L^1(Y,\alpha))$

I'm reading "Anantharaman-Delaroche, C.; Renault, J. Amenable groupoids." (https://mathscinet.ams.org/mathscinet-getitem?mr=1799683), and more specifically the proof Proposition 1.1.5. I will write ...
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Chern classes of $S^2$

It's known that $S^2$ is a $1$-dimensional complex manifold. Let $\varepsilon^n$ denote the trivial vector bundle of rank $n$, then $TS^2\oplus\varepsilon^1 = \varepsilon^3$, so by the Whitney product ...
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1answer
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Relating pullbacks of tautological bundles

Let $X$ be a projective variety and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$ which is generated by its global sections $V=\Gamma(X,\mathcal{E})$. Recall that this gives us a map $f:X\to\...
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1answer
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When are the vector bundles $E$ and $E^\vee \otimes \det(E)$ isomorphic?

Let $C$ be a smooth complex projective curve and let $E$ be a rank two vector bundle over $C$. If $E$ is decomposable, ie $E=L\oplus M$ for some line bundles $L$ and $M$ we have that $$ E^\vee \...
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46 views

Definition of metric on a vector bundle

Let $\xi$ be a real vector bundle over a base space $B$. It is my understanding a metric is meant as a function, $$\beta : E(\xi \oplus \xi) \to \mathbb R$$ where $E$ denotes the total space of the ...
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1answer
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Smooth vector bundles continuously isomorphic are smoothly isomorphic

I have read that if two smooth vector bundles are isomorphic in the category of continuous vector bundles, then they are actually isomorphic in the category of smooth vector bundles. Can someone give ...
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26 views

Smooth structure and vector bundle structure on $L_{alt}^k(TM)$

I want to exhibit smooth structure and vector bundle structure on $L_{alt}^k(TM)=\bigcup_{p\in M} L_{alt}^k(T_pM)$ where $M$ is a manifold of dimension $n$ and $L_{alt}^k(T_pM)$ is the set of all $k$-...
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What is smooth structure on the exterior algebra of cotangent bundle?

Let $M$ be a smooth manifold. Let $(T^*M, \pi, M)$ be cotangent bundle, where $$ T^*M := \bigsqcup_{m\in M} T^*_m M$$ is a set equipped with corresponding topology $O_{T^*M}$ and smooth structure $...
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morphism between pullback bundles

Let $\alpha_1=(E_1,\rho_1,M)$, $\alpha_2=(E_2,\rho_2,M)$ two smooth vector bundles over M and $f:N\longrightarrow M$ a smooth surjective map. If $f^*\alpha_1$ and $f^*\alpha_2$ (the pullback bundles) ...
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Understanding the orientation of vector bundles via double cover

For each real vector bundle of rank $n$, $\zeta: p: V \to X$. An orientation is a continuous choice of orientations of each fiber. Let $\mathbb{R}^{n}$ is given the standard orientation and denote $Or(...
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Intuition behind riemannian metric

I formally understand what a riemannian metric $g$ on a manifold $M$ is. It's basically a section of the vectorbundle $T^*M\otimes T^*M \to M$ (which in the end corresponds to mapping 2 vectors from a ...
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(1,0)-forms/bundle problem

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 69 is this exercise: Can some one give me a hint? I'm kinda new to the subject.
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1answer
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Holomorphic bundle - holomorphic structure problem

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 72 is this theorem: And the proof gose like this: And so on. My question is at the second to last proposition. ...
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$E^* \otimes F = \text{Hom}(E,F)$ for $E,F$ vector bundles

I saw the stement: $E^* \otimes F \cong \text{Hom} (E,F)$ for $E,F$ vector bundles over a manifold $M$, and I want to prove it. My first problem is that i do not know what is the relation $\cong$ here,...
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k-th exterior power bundle $\Lambda^k(M) = \sqcup_{p\in M}\Lambda^k(\tau(M)_p)$ is a manifold

I am trying to prove that for a manifold $M$ with or without boundary of dimension $n$ ,the k-th exterior power bundle $\Lambda^k(M)$ which is defined as disjoint union of the vector spaces $\Lambda^k(...
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Generic condition for vector fields/normal sections

Let $(M,g)$ be a riemannian manifold, and $S\subset M$ a submanifold. I would like to know if there is a result which states that, for some hypothesis about the codimension of $S$, the property for a ...
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1answer
39 views

Horizontal Subbundles and Connection Maps

I'm trying to see the equivalence between the Ehresmann connection and the connection map, and having trouble getting the setup to be correct. Suppose $M$ is a smooth manifold. Let $\pi:TM\to M$ ...
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1answer
39 views

Flat sections of flat vector bundles

Let $E\to X$ be a vector bundle with flat connection $\nabla$. Is there a canonical way to construct a vector subbundle of $E$ from the flat sections ($\nabla \sigma=0$)? More specifically, I would ...
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1answer
41 views

Intuition of a smooth vector field

I read the following statement in pg180, of John Lee's smooth manifold. I am not having much intuition. Let $M$ be a smooth manifold, $X:M \rightarrow TM$ be a rough vector field (i.e. it does not ...
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two versions of the Euler class

I was wondering about cohomological and k-theoretical Euler class, or both versions of characteristic class in general. I mean, one knows that characteristic classes can measure, how twisted such a ...
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Intersection form of projective bundles over $\mathbb P^1$

Let $E$ be a holomorphic line bundle over the complex projective line $\mathbb P^1$. We let denote by $\mathbf 1$ the trivial pundle over $\mathbb P^1$. Motivated by the case of Hirzebrunch surfaces, ...
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1answer
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Restriction of a smooth vector bundle is a smooth bundle?

In John Lee's Smooth manifolds, pg255, he wrote (Restriction of a vector bundle.) Suppose $\pi:E \rightarrow M$ is a rank $k$ vector bundle and $S \subseteq M$ is any subset. We define the ...
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1answer
51 views

Trivialization from a smooth frame

I'm starting to read about smooth vector bundles. My notes suggest that instead of giving a local trivialisation at each point, it is sufficient to give a local frame. I can see that given a rank $r$...
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1answer
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Immersions of vector bundles

Let $E$ be a vector bundle of rank $k$ over a smooth manifold $M$ of dimension $m$. Then $E$ can be thought of as a $(m+k)$-dimensional manifold, so the (weak) Whitney immersion theorem states that $...
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2answers
48 views

Norm topology and compact open topology coincide for vector bundles?

Let $B$ be a compact Hausdorff base space, $p:E \rightarrow B$ a vector bundle. As outlined in Hatcher's, pg 45 we may associate a norm to bundle endomoprhisms, denoted $End(E)$, a norm. Given by $$ ...
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1answer
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Short exact sequence of $K$ functor; applying Tietze extension

I have questions in page 52 of Hatcher's, for the proof of Proposition 2.9 If $X$ is a compact Hausdorff space and $A \subseteq X$ is a closed subspace, then the inclusion and quotient maps $$A \...
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3answers
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Proving continuity of homotopy of clutching functions

This is in page 45 of Hatcher's, that there is a linear homotopy between clutchig functions . I could not workout/justify the linear homotopy. Is there an easy way to see the continuity of homotopy? ...
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Homology of Hirzebruch surfaces

Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\...
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1answer
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Homotopy groups of a projective bundle

I'm learning something about Hirzebrunch surfaces from here. I'm probably missunderstanding something about homotopy groups of these complex surfaces. More precisely, in the link I provided the author ...
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1answer
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Godbillon-Vey class is independent from the choices involved

In the section 2.3 of these notes the Godbillon-Vey class is constructed. It is shown that this class does not depend from the choices involved (lemma 2.11). I have troubles understanding the ...
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How to show that vector field is continuous?

In the book the definition of a a vector field over $U$(open)$ \subseteq S^n$ is given by a continuous map $s: U \to T(U)$ such that $p_U \circ s=id_U$ where $p_U$ is the base point projection from $T(...
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1answer
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inverses in the definition of vector bundle from principal bundle

Pardon my ignorance, but I have a very basic question about the definition of a vector bundle from a principal G-bundle. Let $G$ be a Lie group (I mainly care about the case $G=U(1)$). Let $P \...
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Equivalence of definitions of a vector bundle

Let $n\in\mathbb N$, let $E,B$ be topological spaces and let $p:E\to B$ be a continuous map. For every $b\in B$, let $p^{-1}(b)$ be equipped with the structure of an $n$-dimensional real vector space. ...
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Question on Kahler geometry: Kahler form and $\mathcal{O}_X(1)$

Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $\mathcal{O}_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. ...
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Grassmannian manifold and corresponding vector field

I am assuming the definition of Grassmannian is known. Reference is Vector bundles and K-theory page no 28. I am trying to prove that the map $p:E_n(\mathbb{R}^k)\rightarrow G(\mathbb{R}^k)$ is a ...
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Clutching function for $S^n$: why a vector bundle?

On Page 22 of Hatcher's $K$ Theory we make the following construction For $n,k \in \mathbb N \setminus \{ 0\}$. Let $f:S^{k-1} \rightarrow GL_n(\mathbb R)$ be a continuous map. Define $$...
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the form of classes in $K(X)$ [Atiyah]

Anyone can explain me why taking two bundles $E,F$ we can conclude that any element in $K(X)$ is of the form $E-F$? It is mentioned in Atiyah on the bottom of the page $43$ (I can put screen if needed)...
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1answer
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Degree of curve where matrix of polynomials has rank 1

My question is about a step in Exercise 12.8 on page 442 of 3264 & All That by Eisenbud and Harris. Chapter 12 is about Porteous' formula. The exercise reads: Let $A=(P_{i,j})$ be a $2 \times 3$ ...
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de Rham cohomology of vector bundles: advantages of different definitions of compact vertical cohomology

I'm currently reviewing the Thom isomorphism and the de Rham cohomology of vector bundles over a compact manifold $M$. I'm familiar with the case of topological disk bundles, where the Thom class ...
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1answer
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Existence of an element in $K^0$ group, Koszul complexes

I havee such a question on construction of the Koszul complex (further we are concerned about K-theoretical Euler class). I was thinking of introducing the Koszul complex, and the existing of elements ...
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notation in vector bundles

In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me. We have a map $p:E\rightarrow X$ together with operations $+ : E\times_X E\...
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An Orthonormal Frame $(X_i)_i$ which satisfies $\bigtriangledown _{X_i} X_j =0$ at a point

I am an undergrad student learning Riemannian geometry. My question is about whether you have a nice orthonormal frame in the following sence. Let $(M, g)$ be a Riemannian manifold, with $\...
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$M$ is orientable iff $\bigwedge^n TM \setminus \{ \text{$0$-section} \}$ has $2$ connected components

Let $M$ be an manifold of dimension $n$. We say that $M$ is orientable if it has an $n$-form that doesn't varnishes in any point if $M$. My question is about the following claim: $M$ is orientable ...
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Hatcher's, tensor product of vector bundles : topology explained

So this is a sumamry of the construction given in pg14 of Hatchers of the tensor of two vector bundles: Let $p_1:E _1 \rightarrow B$, and $p_2 : E_2 \rightarrow B$. We define $E_1 \otimes E_2$ to ...
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Direct image of tangent bundle under projection map

Let $\pi: Y \to X$ be a smooth projection map, where $Y$ and $X$ are complex manifolds and the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~~ \forall x \in X$. The ...
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Jacobi fields as “pointing to” nearby geodesics: can we form “triangles”?

Suppose we have two variation vector fields through geodesics, $J_1$ and $J_2$, along a geodesic $\gamma_0$. Then $J_1$ and $J_2$ are Jacobi fields, but I do not think that is imporant for what I am ...
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A compact complex manifold admits an ample line bundle if and only if it is projective

Given a holomorphic line bundle $L$ on a complex manifold $X$, a point $x\in X$ is called a base point of $L$ if $s(x)=0$ for all $s\in H^0(X,L)$ (the space of global holomorphic sections of $L$). The ...
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Example for a Vector Bundle That is Not of Bounded Geometry

Here is the definition of vector bundle of bounded geometry from Shubin: Spectral theory of elliptic operators on non-compact manifolds, page 29: Let $E$ be a complex vector bundle on a manifold $X$. ...