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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30 views

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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How do the Christoffel symbols on an abstract manifold relate to those on submanifolds?

Let $(M,g)$ be a Riemannian manifold of dimension $N$ with (Levi-Civita) connection $\nabla$. I have seen the following definition of Christoffel symbols: For a given smooth moving frame $A=(A_1,\dots,...
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Riemann Manifold equipped with Euclidean metric

I am trying to understand the following equations about the Euclidean metric: $g_p = dx^1 \otimes dx^1 + \dots +dx^n \otimes dx^n$ $X_p\cdot g_p (Y_p,Z_p) = g_p(\bar{\nabla}_{X_p}Y_p,Z_p) + g(Y_p, \...
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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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57 views

Christoffel symbols, dual space

I am confused with the definition of Christoffel symbols for the dual space. Let $M$ be some manifold, $x_i$ local coordinates The Christoffel symbols are defines as $\nabla_{\partial_i} \partial_j = \...
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Positivity of dual Curvature

Let $M$ be a complex manifold or more generally an almorst complex manifold $(M,J)$ with $J \in End(TM)$ such that $J^2=-Id_{TM}$ (if $M$ is complex mfd then it's almost complex al well with $J:=i \...
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66 views

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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66 views

Dimension of the stalk of parallel sections and curvature rank

Let $E\rightarrow M$ be a vector bundle of rank $n$ endowed with a connection $\nabla$. We can define the sheaf of parallel sections by setting $\Gamma_0(U)=\{s\in\Gamma(U):\nabla s=0\}$. Let $\...
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Baby example of a covariant derivative - mistake in Taubes' Differential Geometry?

I'm reading Taubes' Differential Geometry, Section 11: Covariant derivatives and connections. The introduction defines a covariant derivative as an $\Bbb R$-linear map $$\nabla:C^\infty(M,E)\to C^\...
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Parallel transport operator

$\gamma(s)$ is a curve in a four-dimensional manifold. There is an auto parallel on $\gamma(s)$ vector field $\textbf{Y}$ on the manifold. By definition of auto parallel vector, its covariant ...
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Is every compact manifold with a connection geodesically complete?

Let $M$ be a smooth manifold, and let $\nabla$ denote a connection on $M$. Question: If $M$ is compact, is every maximal geodesic of $\nabla$ defined for all $t\in \mathbb{R}$? I know that, if $M$ ...
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Tangent space of a Principal bundle

Suppose we have a Principal bundle with a Lie group G in fiber. It is known that through the trivializations it can locally be expressed as a product of the base space M and the group in the fiber. ...
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Formula for covariant derivatives on principal bundles

I'm reading the notes on gauge theory by José Figueroa-O'Farrill and got stuck on an exercise. To state it, let me first explain my notation. Let $G$ be a Lie group, $P\to M$ a principal $G$-bundle, $...
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How connection helps us to identify two different vector spaces?

I know that section of $TM$ is a vector field on manifold. And I understand that we need a connection to differentiate vectors that live in two different spaces. But still couldn’t grasp the idea how ...
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Calculating number of items in a summed series - non-repeating connections between points

I would like to calculate the number of possible connections in a set of points, and although I can express the idea in a mathematical formula, I don't know how to "reduce" it to a working ...
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Systematic way to find the 1-form of the spin connection?

I'm currently reading Nakahara's book "Geometry, Topology and Physics, 2nd Edition". Section 7.8 discusses Cartan's structure equations, which can be expressed, for a torsion free connection,...
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Connection with torsion

Consider a metric connection $\Gamma^{\mu}_{~~\nu\lambda}$ with torsion $$\Gamma^{\mu}_{~~\nu\lambda} = \tilde{\Gamma}^{\mu}_{~~\nu\lambda}+ K^{\mu}_{~~~\nu\lambda}$$ where $\tilde{\Gamma}^{\mu}_{~~\...
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Euler class of a manifold

While computing the Euler class of the manifold, can we use connections other than the Levi-Civita connection ? What is the restriction on the connection that can be used to compute the Euler class ?
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Holonomy of a flat connection around a contractible loop

I'm currently reading a paper written by John Baez [1]. At a certain point he argues, that the holonomy of a flat connection around a contractible loop is the identity, so its trace in the ...
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Embedding a manifold with connection in $\mathbb{R}^3$ (also need help with diff. eqs.)

Say a universe we want to study has two finite spatial dimensions, which we will imagine to describe a square of side length two. For the diff. manifold underlying the Newtonian spacetime with which ...
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Covariant derivative of a section along a smooth vector field

Suppose $E$ is a $q$-dimensional real vector bundle on a smooth manifold $M$ and $\Gamma(E)$ is the set of smooth sections of $E$ on $M$. A connection on the vector bundle $E$ is a map $$ D:\Gamma(E) \...
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Under which conditions every 2-form is a curvature form?

Let $M$ be a smooth manifold, $G$ a Lie group and $P\rightarrow M$ a smooth principal $G$-bundle. Let $\Omega^1_{eq} (P;\mathfrak{g})$ denote the space of $G$-equivariant $\mathfrak{g}$-valued 1-forms ...
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Chern connection local structure

Let $E$ be a holomorphic vector bundle over a hermitian complex manifold $(X, J,h)$ where $J$ is the complex structure. It is well kown that for every hermitian holomorphic bundle $E \to M$ over a ...
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Connections and second derivatives of curves

If $M$ is a smooth manifold, let $AM$ be the subspace of the double tangent bundle $TTM$ consisting of vectors $v$ such that $\pi^{TM}(v) = \pi^X_*(v)$, where $\pi^X: TX \to X$ is the projection of a ...
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Simplifying the Dirac Equation with a Perturbed Metric

I think I heard it mentioned that one could simplify the Dirac equation by taking the metric to be the perturbation of some simple metric (for example, a perturbation of the Schwarzchild metric): $g_{...
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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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Simple example of non-integrable holomorphic connection

Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves $$\...
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Geometric interpretation of total covariant derivative?

A connection $\nabla$ is said to be compatible with riemannian metric $g$ if $$\nabla_Z g(X,Y)=g(\nabla_Z X,Y) + g(X,\nabla_Z Y).$$ The total covariant derivative $(\nabla_Z g)(X,Y)$ can be ...
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Geodesic-unclear operator $\nabla$

I understand what is a connection $$\nabla_X\textbf{v}$$ for a vector $X$ and a vector field $\textbf{v}$ near point $p$. However I do not understand $(10.5)$ below, namely what is $$\frac{\nabla}{dt}...
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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 ...
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Derivative of real vector fields according to Koszul formula

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 ...
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Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
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Attitude matrix and connection forms

I'm stuck trying to proof this theorem presented at Elementary Differential Geometry O'Neil : Theorem : If $A=(a_{ij})$ is the attitude matrix and $\omega=(\omega_{ij})$ the matrix of connections ...

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