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 $X$.
2,857
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The tensor product of a line bundle with its dual $L\otimes L^*$ is isomorphic to the trivial line bundle
Let $L$ be a holomorphic line bundle and $L^*$ be the dual holomorphic line bundle.
So I believe what we want to show is that the following diagram commutes
(plus some condition on the restriction ...
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On a category of vector bundles. Does it look strange?
Let $X$ be a smooth compact manifold. Let say I want to define a category of vector bundles over $X$, where the objects are of the form $(E, g^E, \nabla^E)$ ($E\to X$ is a complex vector bundle, $g^E$ ...
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Connection with respect to a trivialization
I have been struggling trying to verify the last bit of the following paragraph.
Let $E$ be an arbitary vector bundle on $M$ endowed with a connection $\nabla$. With respect to a trivialization $\psi ...
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Integral Stiefel-Whitney classes and stable almost complex structure
Question: Let $\xi$ be a real vector bundle over a space $B$. Suppose $\xi$ admits stable almost complex structure, i.e., $\xi\oplus
\epsilon^k$ admits almost complex structure for some $k\geq 0$, ...
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Does the direct sum of two real line bundles admit a non-vanishing section?
Let $M$ be a smooth manifold, and $L$ be a smooth real (non-trivial) line bundles over $M$, is it then true that $L\oplus L$ admits a non vanishing section?
Intuitively, I feel like the answer should ...
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Euler class of a principal $SO(2)$-bundle over a lens space
Note that for a manifold $X$, isomorphism classes of principal $SO(2)$-bundles over $X$ are classified by their Euler classes in $H^2(Z;\Bbb Z)$. Now consider a lens space $L(p,q)$; we have $H^2(L(p,q)...
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Existence of principal G-bundle given an associated vector bundle
I am wondering if the following is true.
Let $G$ be a Lie group, $V$ a vector space, $\rho$ a representation of $G$ on V, and $\pi: E\rightarrow M$ a vector bundle with fibre $V$. Does there exist a ...
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Integer times vector bundle notation in Hatcher
In Hatcher's vector bundles and K-theory, p45, there is a proposition written "If $q$ is a polynomial clutching function of degree at most $n$, then $[E,q] \oplus [nE,\mathbb{1}] \approx [(n+1)E,...
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Horizontal and vertical bundles of a linear space
I've been trying to understand the decomposition of the second tangent bundle, $TTQ$, of a smooth Riemannian manifold, $(Q,g)$, into so-called horizontal and vertical bundles, $TTQ = HTQ \oplus VTQ$, ...
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Characterization of Derivations
I am reading the article ON THE BATCHELOR TRIVIALIZATION
OF THE TANGENT SUPERMANIFOLD by O.A Sánchez Valenzuela. Right at the beginning, the following two statements appear:
1.- Let $E\rightarrow M$ ...
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One to one correspondence between metrics on vector bundles and $O(n)$ reductions
During our lecture our professor said that there exists a one to one correspondence between metrics on a vector bundle $E\to M$ and $O(n)$ reductions. He said it is a good exercise to try and prove ...
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Connection $1$-forms and the local expression $d + A$
Let $M$ be a smooth manifold and $E \to M$ a vector bundle over $M$ with a connection $\nabla$. Locally on an open set $U \subset M$ with a frame $(E_1,\dots,E_k)$, we can write any section $s$ of $E|...
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Isomorphism between sections of $\text{Hom}(E,F)$ and bundle maps $E \to F$
Given vector bundles $E$ and $F$ over a smooth manifold $M$ do we have an isomorphism $$\Gamma(\text{Hom}(E,F)) \cong \text{Hom}_{C^\infty(M)}(E,F)?$$
That is, if I have a smooth bundle map $E \to F$ ...
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Obtaining a connection on a trivial bundle by giving a matrix of $1$-forms
I'm new to connections and I'm going over the page (https://mathworld.wolfram.com/VectorBundleConnection.html) in which they state the following
For example, the trivial bundle $E=M\times \Bbb R^k$ ...
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Some problems in the paper about minimal surfaces
I am reading the paper Minimal surfaces of constant curvature in $S^n$, wrote by Professor Robert Bryant in 1985. In this paper, the author introduced some operators to give a new method in order to ...
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Line bundle on projective line
Let $L$ be the (holomorphic) line bundle on $\mathbb{P}^1$ for which the glueing function from $\mathbb{P}^1\setminus \{\infty\}$ to $\mathbb{P}^1\setminus \{0\}$ on the overlap $\mathbb{C}^\times = \...
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Connections on a complex line bundle
Let $L \to M$ be a complex line bundle over a smooth manifold $M$. Let $\{U_\alpha\}$ be a trivializing open cover with transition maps $z_{\alpha\beta}:U_\alpha \cap U_\beta \to \Bbb C^* = \text{GL}(...
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$Z$ definition? (Hussemoller, fibre bundles)
Hussemoller doesn't describe what $Z$ is in this definition (page 67, third edition):
Note that $\mathbf L(\mathbb F^m, \mathbb F^n)$ is the collection of all linear functions $\mathbb F^m \...
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How to show that the map $E \times I \to E: (v,t) \to t\cdot v$ is continuous on a vector bundle $E$?
Let $E$ be a vector bundle over a topological space $M$. I want to show that $\pi:E \to M$ is a homotopy equivalence.
To show this, I use the zero section $\zeta:M \to E$. Then $\pi \circ \zeta = Id_M$...
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Fredholm index of elliptic operator, reference request
Let $E$ be a vector bundle over a smooth manifold $M$ ($M$ may need to be compact and without boundary). Furthermore let$$T:\Gamma(M,E)\to\Gamma(M,E)$$be an elliptic k-th order differential operator. ...
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Affine connection induced by connection map $K: TTM \to TM$
I'm currently reading Riemannian Geometry by Saski, and just finished reading Proposition 4.1 regarding the connection map of a an affine connection. In particular, suppose that $M$ is a smooth ...
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Determinant of endomorphism bundle
We have that the endomorphism bundle of a smooth vector bundle $E$ is $\text{End}(E) = \text{Hom(E,E)}$. Is it necessarily true that $\det(\text{End}(E))$ is always trivial for any smooth vector ...
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Dual bundle transition functions
We can show that the dual bundle $E^*$ with fibers $(E^*)_x = (E_x)^*$ for all $x \in B$ have transition functions $g_{\alpha\beta} = (f_{\alpha\beta}^T)^{-1} : U_\alpha \cap U_\beta \to GL(r,\mathbb ...
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$R^2f_*\mathbb{Z}$ is isomorphic to the constant sheaf $H^2(X_0,\mathbb{Z})$
Let $f:X \rightarrow S$ be a smooth proper family of K3 surfaces and let $X_t=f^{-1}(t)$. Let's suppose $S$ is a disk in $\mathbb{C}^n$.
I know that the Betti/Hodge numbers are constant, so the direct ...
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Computing transition functions from trivial Whitney sum
Suppose $E_1$ and $E_2$ are two vector bundles over a base $B$ such that $E_1 \oplus E_2 \cong \epsilon$, i.e., the sum is trivial. After taking intersections, we get a cover $U_i$ of $B$ such that $...
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Decomposition of the tangent bundle of a tensor product of vector bundles
If $E \to B$ is a smooth vector bundle, then the tangent bundle $T E$ is a vector bundle over both $E$ and $T B$. (If you like, the latter structure is the derivative of the vector bundle structure on ...
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Smooth vector bundles have smooth sections that are not constantly zero
Say $E$ is a smooth vector bundle, then is it necessarily true that $E$ has smooth sections that aren't constantly zero, namely $\Gamma^\infty(E) \neq \{0\}$ for $0$ the zero section of $E$.
My ...
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Splitting bundle quotients
Assume given is a smooth manifold $M$ and smooth vector bundles $A_1\subseteq A_2 \subseteq A_3$ such that $A_i$ is a subbundle of $A_{i+1}$ (embedded submanifold and a vector bundle). Is it generally ...
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Some concrete detail on the Riemann Hilbert correspondence between local system and vector bundle
Let $X$ be a complex smooth manifold. The data of a rank-$n$ local system (a sheaf on $X$ that is locally the constant sheaf $\underline{\mathbb{C}^n}$) is the same thing as an open cover $\{U_\alpha\}...
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Stability of vector bundles as GIT quotient
I believe that the stability of vector bundles or coherent sheaves (defined as the inequality of 'slopes' of its subsheaves) comes naturally from GIT. However, in any literature I can find, it is ...
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Why isn't the third property in the definition of vector bundles redundant?
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological ...
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Tangent space of $RP^n$ vs the orthogonal complement to the line bundle $\gamma_n^1$
I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ ...
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B-Isomorphic? (Husemoller, Fiber Bundles). Illustrated diagrams included.
Wikipedia gives the definition of a bundle map as the arrow $\left( \xrightarrow{\quad \varphi \quad } \right)$ in the commuting diagram below:
Hussemoller gives the definition of a bundle map as the ...
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Is there an isomorphism from a smooth vector bundle $E$ and its double dual bundle $E^{**}$ which doesn't depend on a bundle metric on $E$?
I want to show that the map $F:(E,\pi)\rightarrow (E^{**},\pi^{**})$ define fiberwise to be $F\vert_{E_p}(v)=ev_v$ (where $ev_v:E_p^*\rightarrow\mathbb{R}$ sends $w\mapsto w(v)$ ) is a smooth bundle ...
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Naive linear algebra in vector bundles
I would like to ask a fairly general question: how much of naïve linear algebra remains true for vector bundles? For example,
If $F$ and $F’$ are two subbundles of $E$ such that the fiber $E_x$ is ...
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Analogue of complex projective space, replacing GL1 with GLn
$\mathbb{CP}^1$ can be formed from $\mathbb{C}^\times = \text{GL}_1$ by gluing $\mathbb{C}$ by itself along $\mathbb{C}^\times$, a pushout of $1/z,z : \mathbb{C}^\times \rightarrow \mathbb{C}$.
I am ...
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Does $\mathcal{O}(1)$ on a projective bundle depends on the presentation of vector bundle?
Consider a projective bundle $\pi:\mathbb P(E)\to X$ that comes from a vector bundle of $E$ on a variety $X$. My question is about what is the line bundle "$\mathcal{O}(1)$" on $\mathbb P(E)$...
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Construction of quotient bundles
I have seen a few questions about the constructions of quotients of smooth vector bundles on StackExchange, but I don't think any of them provides a completely satisfactory answer.
I would like to ...
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Computing the reduced $K$-theory of $\mathbb CP^n$ using AHSS
I'm studying Atiyah-Hirzebruch spectral sequence from some slides, a bit hard to follow as they are a collage of pages from various books and papers, filled with writings and deletions made by hand by ...
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Details of the proof that $T \mathbb{R}P^n \cong \operatorname{Hom}(\gamma_n, \gamma_n^\perp)$
Let $\gamma_n = \{(\ell, v) \in \mathbb{R}P^n \times \mathbb{R}^{n + 1} \mid v \in \ell\}$ be the tautological line bundle over $\mathbb{R}P^n$, and let $\gamma_n^\perp$ be its orthogonal complement ...
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A question regarding linearized line bundles on a variety with a group action
Let $G$ be an affine algebraic group, and let $X$ be a smooth variety with a $G$-action $\sigma:G\times X\rightarrow X$. Let $\pi: G\times X \rightarrow X$ denote the second projection. A $G$-...
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Are function spaces over a shrinking set vector bundles?
I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the ...
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Continuity of a map into $G_n(\mathbb R^N)$
Let $\xi:E\to X$ be a $n$-dimensional vector bundle embedded in the trivial bundle $X\times \mathbb R^N\to X$, for some $N\ge n$. Denote also by $\tau:O\to G_n$ the tautological bundle, where $G_n:=...
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Technique to prove that certain quasi-vector bundles are trivializable
Fixed a topological space $S$, let $X\subset S\times \mathbb R^n$ be a subspace, with respect to the product topology, such that $(\{s\}\times \mathbb R^n)\cap X$ is a subspace of the vector space $\{...
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Lifting a duality pairing to a vector bundle morphism
Let $ M $ be a smooth manifold. Given $ p\in M $ denote with
$$
\langle{-},{-}\rangle_p\colon \mathrm T_p^*M\otimes \mathrm T_pM\to \mathbb R
$$
the canonical duality pairing between the tangent space ...
3
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1
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Questions on notation that I haven't seen before (tensor product sign in superscript and mysterious $\Gamma$ symbol)
I have some questions about the Wikipedia article on Tensor fields. The definition of a tensor field is given as such:
Let $\mathfrak M$ be a manifold, for instance the Euclidean plane $\mathbb R^n$.
...
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Proof criticize: rank $n$ vector bundle on $\mathbb{A}^1_k$ is trivial
The problem stated in the title comes from Vakil's AG notes 14.3.C. I am aware that there are few similar questions answered here. But I wish to give my own proof that I am not so sure if is valid. ...
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1
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Nice example for a splitting short exact sequence of vector bundles [closed]
I am preparing an introduction to banach manifolds and currently I am working on exact sequences of vector bundles of banach manifolds. Are there any nice examples (i.e. easy to understand for people ...
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104
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Vanishing of connection matrices for flat principal $G$-bundle
Background
Recall that for a real vector bundle, there is a well known integrability theorem.
Theorem. Suppose there is a vector bundle $E$ with fiber $\mathbb R^n$. If $A$ is a flat connection on $E$,...
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Under what geometric conditions is the image of an irrep of $H$ isomorphic to the image of an irrep of the fundamental group of $G/H$?
Let $G$ be a Lie group and $H$ a closed subgroup. Let $\pi_1(G/H)$ denote the fundamental group of the homogeneous space $G/H$. If $\lambda$ is an irrep of $H$, then under what conditions does there ...