# Tagged Questions

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### Tangent bundle of a tubular neighborhood

Let $N \to X$ be normal bundle of a submanifold $X$ of $Y$. How can I prove that $TN|_{TX}$ is isomorphic to the normal bundle of the inclusion $TX\to TY$? And why this vector bundle is isomorphic to ...
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### Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
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### Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
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### R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
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### Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$\phi: M \to N.$$ As far as I understand, this gives rise to two distinct isomorphisms $$a : \mathcal A^1(M) \cong \mathcal A^1(N) :b$$ between the space of ...
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### The difference between a fiber and a section of a vector bundle

If $E_x := \pi^{-1}(x)$ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E$ with $\pi \circ s = id_M$. This implies that $\pi^{-1} = s$ on $M$. So then ...
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### How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
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### Are vector bundles just modules over $C^{\infty}(M)$, or are “locality” conditions required?

In this question the asker defines 1-forms on a (real, smooth) manifold $M$ to be $C^{\infty}(M)$-module homomorphism[s] from $Vec(M)$ to $C^{\infty}(M).\:\:\:\:(*)$ I'm wondering if this is ...
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### Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
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### Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
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### Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0$$ Here $VE$ is ...
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### Prove the existence (or well-definedness) of the induced connection in tensor bundle

Given a connection $\nabla$ on a vector bundle $E$ over a smooth manifold $M$, we know there is a unique extension of $\nabla$ to all tensor bundles of $E$ that satisfies Leibniz rule and contraction. ...
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### Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
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### Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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### Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
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### Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...
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### Recover Covariant Derivative from Parallel Transport

It is well known that one can recover the connection from the parallel transport. I struggle to understand this concept. Since $\Gamma(\gamma)^t_s:E_{\gamma(s)}\to E_{\gamma(t)}$ is an isomorphism ...
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### Tensorproduct of vectorbundles

Assume we got $\pi:E \to M$, a vector bundle of manifolds. I know how to make the bundle $(E \otimes E)^* \to M$. But how do the local trivializations look like ? I suppose that if ...
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### Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?

Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts ...
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### Unitary connection and Hermitian connection

I am confused with the two notions. I basically understand Hermitian connection: it is a complex analog of metric-compatible connection on $TM$, a connection that preserves the hermitian metric $h$ ...
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### The tangent bundle over a manifold is trivial iff the manifold is paralelizable

Why is the tangent bundle over a manifold trivial if and only if the manifold is parallelizable? What additional condition do we need to impose on a fiber bundle if we want it to be trivial exactly ...
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### The tangent bundle over a manifold is locally trivial

Assume $M$ is an $m$-dimensional manifold and $(U,h)$ is a chart. Denote the differential of $h$ at the point $p$ by $T_ph$. How can I verify that $pr_1(Th(p,x))=h(p)$ where $pr_1$ is the projection ...
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### Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
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### Vanishing of non top-order Chern classes

Let $E \to B$ be a rank-$r$ complex vector bundle and denote by $c_1(E)$, $\ldots$, $c_r(E)$ its Chern classes. Then $c_r(E)$ is just the Euler class of the realization of $E$ as a real vector bundle ...
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### vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
I've concluded a differential geometry course (mainly covering classical results about parametric surfaces or diff. surfaces in $\mathbb{R}^3$, for example Gauss' Theorema Egregium or geodetics) which ...