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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Role of connection in a differentiable manifold

It's almost clear that a linear connection can be viewed as Vector bundle that makes a PARALLEL TRANSPORT .But I don't understand how they mean to connect vectors. What is the role of connection in ...
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43 views

A diffeomorphism which maps geodesics to geodesics preserves the connection?

Let $(M,\nabla^M),(N,\nabla^N)$ be two smooth manifolds with given (affine) connections on their (tangent bundles). We say a diffeomorphism ,$\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ is an ...
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38 views

Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
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2answers
49 views

Parallel transport along a 2-sphere.

I'm currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below. From my ...
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23 views

Difference between types of connections [closed]

For my background, I am familiar with the basics of differential geometry, especially Riemannian geometry, and in some more advanced topics relevant to physics, especially general relativity. Lately ...
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34 views

Looking for a reference that explains connections and curvature by double tangent space

I'm looking for a book or a set of lecture notes on differential manifolds that explain connections (Levi-Cevita connection, prinicipal connections) and curvature on an abstract manifold from the ...
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41 views

If $\nabla_1$ and $\nabla_2$ are Levi-Civita connections for a metric on the smooth sphere, then their curvature tensor would recover the radius…?

I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths. Let ...
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14 views

Difficulty proving gauge invariance on an SU(N)-valued potential

Say we have a four-dimensional spherically symmetric $\mathfrak{su}(N)$ gauge potential in standard Schwarzschild co-ordinates which can be written \begin{equation} ...
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41 views

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
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49 views

Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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1answer
46 views

Question of well-definedness of the Levi-Civita connection?

On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, ...
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42 views

Connection giving an identification of the double-tangent bundle

Let $M$ be a manifold, and let $\nabla$ be a (possibly torsion-free) connection on $M$. Then I am pretty sure that $\nabla$ induces an isomorphism between the double-tangent bundle and the Whitney sum ...
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90 views

Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
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38 views

Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
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73 views

Is this operation meaningful or it is a mistake in the book?

I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is ...
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52 views

Manifold,affine connection,vector field

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the ...
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30 views

Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
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52 views

Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...
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1answer
69 views

Exercise about Christoffel symbols: prove that $\Gamma^k_{ij}(p)+\Gamma^k_{ji}(p)=0$

Let $\nabla$ be a linear connection on a manifold $M$. I want to prove that: $$\Gamma^k_{ij}(p)+\Gamma^k_{ji}(p)=0,$$ if in $p\in M$ normal local coordinates are defined. I have a solution which ...
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Let be $K$ a tensor field, what is the relation between the coefficients of $K\big|_U$ and $\nabla K\big|_U$?

Let be $\nabla$ a linear connection on a manifold $M$ and $K$ a tensor field of type $(h,k)$. Using a local coordinate system $(U, \phi)$ we can write: $$ K\big|_U= \sum ...
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76 views

Introductions to Ehresmann connections and Chern-Simons forms

I am looking for introductory texts on Ehresmann connections and Chern-Simons forms. I seek detailed, hands-on presentation. Please, recommend sources that employ a differential forms approach rather ...
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26 views

Confusion on when components of a vector relative to a basis are not components of a tensor

I have been studying affine connections, parallel transport and the covariant derivative. The text I am reading defines an affine connection $\nabla$ as a map $\nabla ...
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72 views

commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
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How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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68 views

When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
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37 views

are extensions of flat connections flat?

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...
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33 views

Quick question about covariant derivative

Let $f$ be a function and define $\nabla_X f = X(f)\,\,(1)$, where $\nabla$ is the connection on a manifold and as far as I understand the r.h.s is a function and $X$ is a vector field. I am just a ...
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28 views

What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
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Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
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119 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
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59 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
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58 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
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119 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
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Covariant derivative: The meaning

Many text books on differential geometry motivate covariant derivative more or less by saying that if you have a vector field along a curve on a manifold (that is a curve $\gamma(t)$ and an assignment ...
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71 views

Two definitions of curvature

my question is about the compatibility of two definitions of curvature of a Riemannian manifold. In particular I refer to the one from algebraic geometry and the one from differential geometry. ...
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48 views

Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
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61 views

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
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75 views

Deriving Ricci identity for co-vector fields

Let $\nabla$ be the covariant derivative associated with a torsionless connection. Prove the Ricci identity for covectors: $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = ...
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2answers
72 views

Gauge fields and restrictions of the connection one form

I am working through some lecture notes on principal bundles and am stuck on the proof of a certain proposition. In the following, $\pi:P\rightarrow M$ is a principal bundle, $\omega$ is the ...
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1answer
54 views

Ask for an explicit proof of linear lemma in Riemannian geometry

Lemma Let $f:(M,g)\rightarrow(\bar{M},\bar{g})$ be an isometry between two Riemannian manifolds, then $df(\nabla_X Y)=\bar{\nabla}_{df(X)} df(Y)$ where $\nabla,\bar{\nabla}$ are Riemannian connections ...
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1answer
50 views

Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
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Connections and tensor fields

Let $T$ be a $(1, 1)$ tensor field, $\lambda$ a covector field and $X, Y$ vector fields. We may define $\nabla_X T$ by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = ...
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Preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
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60 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
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51 views

Failure to close of horizontal lift on principal bundles.

When considering the geometrical meaning of the curvature on principal bundle $\pi: P \rightarrow M$, let us consider a coordinate system ${x_μ}$ on a chart $U$. Let $V = ∂/∂_{x_1}$ and $W = ...
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115 views

Existence of a holomorphic connection implies existence of a flat connection

Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local ...
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Induced connection is well-defined

Let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, then $$\forall ...