# Questions tagged [connections]

In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. (Def: http://en.m.wikipedia.org/wiki/Connection_(vector_bundle))

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### do Carmo Riemannian Geometry Exercise 2.3: definition of $\nabla$ for an immersion - Part II

The following is Exercise 3 of Chapter 2 of my Brazilian edition of do Carmo's Riemannian Geometry: Let $f: M^n \to \overline M^{n + k}$ be an immersion from a differentiable manifold $M$ to a ...
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### do Carmo Riemannian Geometry Exercise 2.3: definition of $\nabla$ for an immersion

The following is Exercise 3 of Chapter 2 of my Brazilian edition of do Carmo's Riemannian Geometry: Let $f: M^n \to \overline M^{n + k}$ be an immersion from a differentiable manifold $M$ to a ...
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### Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
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### “Pedantic” derivation of geodesic equation using pullback bundles

I'm trying to get more comfortable with manipulations involving connections and vector fields so I've tried to derive the geodesic equations without having to resort to any familiarities using ...
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### Do we need gammas to determine $\nabla$?

I know that something must be wrong with the following calculation - otherwise, the covariant derivative could be defined intrinsically on a differentiable manifold - but I don't seem to be able to ...
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### Is there a proof via Chern-Weil theory that the first Chern numbers of two $\mathbb R$-isomorphic complex vector bundles are equal mod 2?

Let $M$ be a compact, oriented surface. If we have a complex vector bundle $E$ over $M$, then we can define the first Chern number $c_1(E)$ via Chern-Weil theory. More precisely, if $\nabla$ is a ...
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### Robertson Walker metric, covariant derivative

I am currently trying to understand the paper "Global Wave Maps on Robertson–Walker Spacetimes" by YVONNE CHOQUET-BRUHAT and don't understand the following part where this $D$ is defined. The context ...
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### When is a symplectic connection a symplectomorphism?

Let $(M, \omega)$ be a symplectic manifold and $\nabla$ a symplectic connection, i.e. an element of $\Omega^1 (M, \operatorname{End} (TM)).$ Denote by $\operatorname{End}_{\omega} (TM)$ a bundle of ...
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### Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
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### Are Christoffel symbols associated with a tensor object?

First of all, my question lies on: Differentiable, real, n-dimensional Manifolds and in the context of differential geometry for General Relativity. Also, my level of academic mathematical language do ...
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### Levi-Civita-Connection on $2$-Form

Let $(M, g)$ be a Riemann manifold and $$\nabla^{LC}: \Gamma(M,TM) \to \Gamma(M, T^*M \otimes TM)$$ the Levi-Civita connection over the tangential bundle $p:TM \to M$. Since in general for arbitrary ...
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### Relation between the differential of a vector field and the covariant derivative

Let $\mathcal{M}$ be a Riemannian manifold and let $X:\mathcal{M} \rightarrow \mathcal{T}\mathcal{M}$ be a smooth vector field, i.e., $X(p) \in \mathcal{T}_p\mathcal{M}$ $\forall p \in \mathcal{M}$. ...
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### Two linear connections have the same geodesics if their difference tensor is antisymmetric

Given two linear connections $\triangledown^a$ and $\triangledown^b$ on a manifold M, their assosciated difference tensor is $D(X,Y)=\triangledown^a_XY-\triangledown^b_XY$. I have confidently verified ...
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### Exact differentiation

How it follows below after $9.4$ that "...and so $d\sigma^i=0$ for all $i$" ?
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### Curvature tensors and local isometries (Homework)

I am asked first to to prove that an isometry ($\Phi$) preserves the Levi-Civita connection, this is: $$\Phi_*(\nabla_XY)=\nabla^*_{\Phi_*X}\Phi_*Y$$ where $\nabla^*$ refers to the Levi-Civita ...
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### Clarification on the auto parallel equation derivation

I've been following the lectures from the winter school on gravity and light, which has some (I think) a good introduction to Differential Geometry. However, in one of the lectures, there is a certain ...
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### Two different interpretations of connections over vector bundles as an affine space

The following is not exactly a question but a consideration that needs to be completed, and I wish such a completion can be provided as an answer to this post. I am writing here such result of a ...
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### Sufficient condition on extending a vector to a parallel vector field.

Extending a tangent vector to a parallel vector field Say we have a vector $Z \in T_pM$ for Riemannian manifold $(M,g)$. Lets consider a possible method for extending this vector to a parallel vector ...
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### Vector bundle with flat connection over simply connected manifold is trivial

I am trying to finish a proof of the statement in the title. Let $M$ be a simply connected smooth manifold and $E \twoheadrightarrow M$ be a vector bundle with a flat connection. Since the bundle is ...
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### Category of vector bundles with connections

Vector bundles with connections over the same manifold $M$ make up a category. Indeed, let $E, E' \twoheadrightarrow M$ be vector bundles with connections $\nabla$ and $\nabla'$. A morphism between ...
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### $\nabla_a \nabla_b v^c$ abstract index notation

I am having some trouble with the equivalence between abstract index notation (AIN) and standard tensor component notation (TCN, for short). Let us consider a covariant derivative $\nabla$. In TCN, ...
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### Hitchin fibration and twisted connections

Let $p$ be a positive prime. Let $X,Y$ be smooth projective curves over a scheme $S$ of characteristic $p$, and let $\pi:X\times_S Y \to X$ be the first projection. Let $\mathcal E$ be a vector ...
I'm trying to understand covariant derivative in the simplest case of $\mathbb{R}^n$ here. On a Riemannian manifold $(M,g)$, for a connection $\nabla$ which is compatible and torsion-free, I have ...