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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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Every line bundle on a complex algebraic curve has a meromorphic section

Every line bundle $L$ on a complex algebraic curve $X$ is of the form $\mathcal{O}(D)$, where $D$ is some divisor on $X$. This means $L$ has at least one nonzero meromorphic global section, i.e. $$H^...
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Tropical vector bundles?

Is there a nice notion of tropical vector bundles? A cursory google turns up https://arxiv.org/abs/0911.2909 but it is unclear to me how these objects are like usual vector bundles.
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Constructing the flat vector bundle associated to a given linear representation of the fundamental group

I'm reading this notes, and I have some questions about the contruction on page 18. Let $M$ be a connected manifold and $E\rightarrow M$ a flat vector bundle over $M$. Consider $\{(U_\alpha, \phi_{...
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Order in a family of elliptic differential operators is constant

In reading the book Spin Geometry by Lawson and Michelsohn I found this definition of a continuous family of elliptic operators: Definition. Let $E,F$ be smooth vector bundles over a smooth ...
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How i x j= k (vector) BUT ixj = (i)(j) (sin90) = (1)(1)(1) = 1 (Scalar) [closed]

How i x j = k (vector) , also in josiah willard Gibbs book who first given the idea of cross product did not explain the mathematical way of cross product. Also from quaternions i found no real ...
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Two definitions of a connection

For me a connection $\nabla$ on a vectorbundle $E$ over a smooth manifold $M$ is a $\mathbb{R}$-bilinear map $\Gamma(TM)\times\Gamma(E)\rightarrow\Gamma(E)$ which is tensorial in the first slot and ...
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Let $B$ be a vector bundle over a compact metric space $X$. Is there always a linear bundle automorphism that covers $f \in \mbox{Homeo}(X)$?

Let $B$ be a vector bundle over a compact metric space $X$. Is there always a linear bundle automorphism that covers $f \in \mbox{Homeo}(X)$? By covering I mean that $fp=pF$, where $p: B \rightarrow ...
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Does existence of a parallel section on $E$ imply local existence of a parallel section on $\bigwedge^k E$?

Let $E$ be a smooth vector bundle equipped with an affine connection $\nabla$. Suppose that $(E,\nabla)$ admits a non-zero parallel section. I think that $(\bigwedge^k E,\bigwedge^k \nabla)$ does ...
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Vector bundle over linear maps with fixed rank

Let $V_1, V_2, W_1, W_2$ be finite dimensional vector spaces over an algebraically closed $K$ field and $r_1, r_2$ positive integers. For $i \in \{1,2\}$, let $X_i$ be the set of $K$-linear maps $W_i \...
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1-form sufficiently $C^1$-close to the zero section

Question: If $\mu$ is a 1-form sufficiently $C^1$-close to the zero 1-form, then $$\{(p, \mu_p) ; p \in M, \mu_p \in T_p^*M \}\cong\text{Graph}\ f$$ for some diffeomorphism $f: M \to M$. ...
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General tips for showing that subspaces “vary continuously”

Here is (part of) a problem from Spivak's Differential Geometry Vol. 1. 6. For a bundle map $(\tilde f,f),$ with $f:B_1\to B_2,$ let $K_p$ be the kernal of the map $\tilde f|_{\pi_1^{-1}(p)}$ from $...
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Non-conformal metrics on vector bundles where $\nabla g=\omega(\cdot) g$

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M \ge 2$), equipped with a metric $g$ and a connection $\nabla$, such that $\nabla_X g=\omega (X) g$ for every vector field $X$ on $M$. ($\...
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A Topological Invariant for $\pi_3(U(n))$

Recently, I saw a construction of topological invariant for $\pi_3(U(n))$ with $n\geq 2$: $$ N=\frac{1}{24\pi^2}\int_{S^3} d^3x\ \epsilon^{ijk} Tr[(U^{-1}\partial_{x_i}U)(U^{-1}\partial_{x_j}U)(U^{-1}\...
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An isomorphism of vector bundles over a manifold, $K(X)$,

Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that $$E_1 - E_0 =0 \in K(X)$$ (I believe we also suppose $X$ to be compact so $K$-theory makes sense here.) Proposition: If $\...
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Extending a monomorphism of bundles, Lemma 7.3, Atiyah, Shapiro

This is from Lemma 7.2, pg17 Let $E,F$ be vector bundles on $X$ and $f:E \rightarrow F$ a monomorphism on $Y$. Then if $\dim F > \dim E+ \dim X$, $f$ can be extended to a monomorphismon on $X$ ...
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Why is the $C^\infty(M)$-module of smooth sections of a vector bundle $E$ not free?

Let $M$ be a second countable smooth manifold. When I learned about differential geometry, a side note was made about how if $E$ is a vector bundle, $\Gamma(E)$ is a $C^\infty(M)$-Module that is not ...
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Bundle of Endomorphism of Line Bundle always trivial

Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact). Let $L$ be a line bundle over $B$. My question is how to see that the morphism ...
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Operator K-theory and Topological K-theory

I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_\infty (...
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Given a differential form $\omega$, is there a differential form $\phi$ such that $\omega\wedge\phi$ is closed?

Let $M$ be a differential manifold and $\Omega^p(M)$ the vector bundle of $p$-forms. My question is: Given a differential $p$-form $\omega$, is there a differential $q$-form $\phi$ such that $d(\...
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Locally trivial line bundle

I am reading the book "An Introduction to Contact Topology" by Geiges. In the proof of Lemma 1.1 where he proves that any co-dimension 1 hyperplane field distribution is locally kernel of a 1-form, ...
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Existence of connection on dual bundle

I quote the construction given in Madsen's Calculus to Cohomology. This is more or less the construction explained here defining connection on dual bundle. For a vector bundle $\Omega^i(\xi) := \...
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Are there topologically trivial bundles with a nonzero curvature?

A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder. Is it possible to have a ...
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Isomorphism of bundles , Madsen's Calculus to Cohomology

Let $\xi=(E,M,p)$ be a smooth bundle over $M$. Denote $\Omega^0(\xi)$ the smooth sections. We also define $\Omega^i(M):= \Omega^0(\wedge^i T^*M)$, differential $i$ forms, and $\Omega^0(M)$ is then ...
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A “natural homomorphism” $H^n(B; \Bbb Z) \rightarrow H^n(B; \Bbb Z_2)$.

This is Proposition 4.12, pg33 The claim of the statement is: The natural homomorphism $\Gamma:H^n(B;\Bbb Z) \rightarrow H^n(B ; \Bbb Z_2)$ sends the Euler class to the top Stiefel Whitney class. ...
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Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $...
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Pushforward of a line bundle along a finite morphism of curves

Let $f:X\rightarrow Y$ be a finite morphism (a branched covering) of degree $n$ of smooth complex algebraic curves. It is a known result that for any line bundle $L$ on $X$, the pushforward $f_* L$ ...
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A theorem on fibre bundles, $H^k(E) \cong H^{k+n}(E,E_0)$.

This is a theorem about fibre bundles, stated in page 21, Theorem 3.6 Let $\xi= (E,B,p)$ be an $n$-vector bundle. Let $F_0$ be the fibre $F$ without its nonzero element, and $E_0$ be the total ...
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Well-posedness of linear ODE problem on vector bundle

Let $\gamma \colon [0,\alpha] \to \mathbb{R}^{3}$ be a smooth regular curve and let $N\gamma$ denote its normal bundle. Recall that $N\gamma$ is a smooth vector bundle, whose fiber at $\gamma(t)$ is ...
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Showing a map $f:E \rightarrow \Bbb R^\infty$ is continuous.

Suppose for $i \in \Bbb N$, we have continuous maps $g_i:E \rightarrow \Bbb R^n$ satisfying local finiteness proeprty i.e. for each $x\in E$ exists nhood $U_x$ such that only finitely many $g_i$ are ...
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Can we ignore the ``holomorphic trivialisation'' in the definition of a holomorphic vector bundle?

I have learnt two definitions about holomorphic vector bundles over a complex manifold $M$. $E\to M$ is a smooth complex vector bundle with a trivialisation such that the transition functions are ...
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Continuity of a map: proving manifold structure of Grassmanian

This post should be self contained - it is from a concern of page 13 of this notes. Let $q \ge n$, $V_n(\Bbb R^q)$ be the subspace of $n$ -linearly independent vectors in $(\Bbb R^q)^n$. We ...
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Dimension of kernel of a operator

This question is simply applying a theorem. But I do not understand how . One can treat most of the content as black box. I will provide the definitions. The context: I want to show If $M$ is a ...
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Orientation on normal bundle

Let $M \subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17: this induces an orientation on the normal bundle of $M$.
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Smooth quotient bundle

Let $E \rightarrow M$ be a smooth vector bundle. This link gives a construction of quotient bundle for a subbundle. $E' \subseteq E$. We define an equivalence relation $\sim$ on $E$ by $v_x \sim ...
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An exact sequence of vector bundles.

Page 376, Prop 15.6.7: Let $p:E \rightarrow M$ be a vector bundle. There exists a canonical exact sequence $$ 0 \rightarrow p^*E \xrightarrow{\alpha} TE \xrightarrow{\beta} p^*TM\rightarrow 0 $$ ...
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A smooth vector bundle $p:E \rightarrow B$ is a submersion

Prove that a smooth vector bundle $p:E \rightarrow B$ is a submersion. With $B$ being a smooth $k$ dimensional manifold, and $E$ be a bundle of dimension $n$. I would like to see how one prove this. ...
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Projection to subbundle is continuous in vector bundle

Definition: A subbundle $W \subseteq V$ of vector bundle $V \rightarrow X$ is a union vector subspaces $\bigsqcup_x W_x$ which is locally trivial with the subspace topology. Lemma: A subbdunel $...
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Connection matrix in orientable 2-bundle is skew-symmetric

In the notes I am following to learn about connections, there is the following lemma: whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric ...
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Forms on vector bundles: vertically compactly supported

Definition: Let $\pi:V \rightarrow M$ be a vector bundle. $\Omega^p_{cv}(V)$ is the sections of $p$ form on $V$, such that $\pi^{-1}(K) \cap supp \, (w) $ for all $K \subseteq M$ compact. ...
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Geodesics in $\mathbb{R}^n$ with the trivial connection

Define geodesic as follows: Given a tangent bundle $TM\rightarrow M$ with connection $\nabla$, a geodesic is a curve $\gamma:I\rightarrow M$ such that $(\gamma^*\nabla)\dot \gamma = 0$. (Notice ...
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Covariant Exterior Derivative action on the $\mathrm{End}(E)$-valued p-forms

Suppose I define an operator $d_A$ by its action on sections $s\in \Gamma(E)=\Omega_M^0(E)$ of some vector bundle $\Pi:E\rightarrow M$ in a trivializing neighbourhood $U\subset M$ as $$ d_As|_U=(ds+A\...
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Given a map $X \to \text{GL}_2(\mathbb{R})$ how do I determine a flat connection on this Riemann surface?

I need help determining the Euler class of this vector bundle $\phi:E\to X$. The base space is the torus $X = \mathbb{R}^2/\mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) \simeq \mathbb{R}^2$....
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Use of Leibniz rule in a proof envolving a metric connection in a vector bundle

Let $E\rightarrow M$ be a vector bundle with metric $g$ and metric connection $\nabla:\Omega^0(M,E)\rightarrow \Omega^1(M,E)$. I am trying to understood this short proof: I do not understand the ...
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Vector bundles are reps of $\Gamma_\mathbb{R}$

It's well-known that every flat holomorphic vector bundle $V$ of rank $n$ on complex manifold $X$ comes from a representation $$\rho\ : \ \pi_1(X) \ \longrightarrow \ \text{GL}(n,\mathbb{C})$$ by ...
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Infinitely many fixed-rank bundles on projective space?

I was asked as a question whether a topological space can have infinitely many non-isomorphic bundles (with fixed rank). As a hint we were recommended to consider projective space. I don't really know ...
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When is a vector bundle morphism a vector bundle?

When does a morphism $f:E'\to E$ of vector bundles over the same base $B$ make $E'$ a vector bundle over $E$? Definitely $f$ has to be surjective, and this seems like it is also sufficient. Because ...
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Existence of Thom Class

In page 133, Theorem 8.5.5. (The Thom isomoprhism theorem) Let $\pi:V \rightarrow X$ be a complex vector bundle of rank $n$, over al locally comapct space $X$. Let $$ 0 \rightarrow \pi^* \...
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Quasicoherent sheaves on the groupoid of vector bundles on a surface

Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$\mathcal L\in QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a ...
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Relative $K$-theory, definition

On pg125 on proving that that the concordance classes quotiented by the acyclic ones froms a group, there is a lemma: Lemma 8.4.5 Let $(E,F,f)$ and $(E,F,g)$ be two $K$-cycles on $(X,Y)$. Assume ...
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Topology of decomposition of a space

In pg 121 of this notes, the author outlines a construction of gluing bundles. The scenario begins with Let $X= X_0 \cup X_1$ be union of two comapct spaces. $A = X_0 \cap X_1$ so that $X = X_0 \...