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 $X$.
1,518 questions
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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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26 views
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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17 views
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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0answers
32 views
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
27 views
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
39 views
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
$$
E^\vee \...
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1answer
46 views
Definition of metric on a vector bundle
Let $\xi$ be a real vector bundle over a base space $B$. It is my understanding a metric is meant as a function,
$$\beta : E(\xi \oplus \xi) \to \mathbb R$$
where $E$ denotes the total space of the ...
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1answer
15 views
Smooth vector bundles continuously isomorphic are smoothly isomorphic
I have read that if two smooth vector bundles are isomorphic in the category of continuous vector bundles, then they are actually isomorphic in the category of smooth vector bundles.
Can someone give ...
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0answers
26 views
Smooth structure and vector bundle structure on $L_{alt}^k(TM)$
I want to exhibit smooth structure and vector bundle structure on $L_{alt}^k(TM)=\bigcup_{p\in M} L_{alt}^k(T_pM)$ where $M$ is a manifold of dimension $n$ and $L_{alt}^k(T_pM)$ is the set of all $k$-...
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23 views
What is smooth structure on the exterior algebra of cotangent bundle?
Let $M$ be a smooth manifold. Let $(T^*M, \pi, M)$ be cotangent bundle, where
$$ 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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1answer
17 views
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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1answer
17 views
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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2answers
45 views
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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0answers
38 views
(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
18 views
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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0answers
16 views
$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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0answers
46 views
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(...
3
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0answers
99 views
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 ...
1
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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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0answers
14 views
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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0answers
13 views
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, ...
3
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1answer
88 views
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$...
4
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1answer
80 views
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 $$ ...
1
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1answer
45 views
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
55 views
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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0answers
51 views
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
32 views
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 ...
2
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1answer
67 views
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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0answers
41 views
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
15 views
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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0answers
48 views
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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1answer
34 views
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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1answer
34 views
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 ...
3
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2answers
67 views
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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0answers
37 views
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
35 views
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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0answers
31 views
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
38 views
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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2answers
33 views
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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0answers
22 views
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 $\...
3
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1answer
61 views
$M$ 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 ...
2
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1answer
76 views
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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0answers
111 views
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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0answers
17 views
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 ...
4
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0answers
59 views
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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0answers
23 views
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 $X$.
...
