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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: https://en.wikipedia.org/wiki/Connection_(vector_bundle))

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. Reference: Wikipedia.

If the fiber bundle is a vector bundle, then the notion of parallel transport must be linear. Such a connection is equivalently specified by a covariant derivative, which is an operator that can differentiate sections of that bundle along tangent directions in the base manifold.

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Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics, he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I have absorbed the main intuition behind each element…
Zev Chonoles
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Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following conditions: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) Symmetricity, i.e.…
evgeny
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Does the Levi-Civita connection determine the metric?

Can I reconstruct a Riemannian metric out of its Levi-Civita connection? In other words: Given two Riemannian metrics $g$ and $h$ on a manifold $M$ with the same Levi-Civita connection, can I conclude that $g=h$ up to scalars? If not, what can I say…
archipelago
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What is the affine connection, and what is the intuition behind/for affine connection?

Here is the definition of affine connection, as appears in Milnor's book Morse Theory. DEFINITION. An affine connection at a point $p \in \text{M}$ is a function which assigns to each tangent vector $\text{X}_p \in \text{TM}_p$ and to each vector…
user416548
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Collecting math websites

I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and…
user657166
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Prove that Christoffel symbols transformation law via the metric tensor

It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is: $$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over…
Arthur
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Torsion of a connection as an obstruction to integrability

Let $E$ be a vector bundle over a smooth manifold $M$ equipped with a linear connection $\nabla : \Gamma(E) \to \Omega^1(M;E).$ I say $(M,E,\nabla)$ is flat if it admits trivial local models; i.e. if for each $p \in M$ there is a $\nabla$-parallel…
Anthony Carapetis
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Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = \int \langle \alpha, \delta \beta \rangle…
Yrogirg
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Recovering connection from parallel transport

(doCarmo, Riemannian Geometry, p.56, Q2) I want to prove that the Levi-Civita connection $\nabla$ is given by $$ (\nabla_X Y)(p) = \frac{d}{dt} \Big(P_{c,t_0,t}^{-1}(Y(c(t)) \Big) \Big|_{t=t_0}, $$ where $p \in M$, $c \colon I \to M$ is an integral…
Andrew Whelan
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Understanding the notion of a connection and covariant derivative

I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative. As…
Will
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Covariant derivative of vector field along itself: $\nabla_X X$

Consider a vector field $X$ on a smooth pseudo-Riemannian manifold $M$. Let $\nabla$ denote the Levi-Civita connection of $M$. Under which conditions can something interesting be said about the covariant derivative of $X$ along itself, i.e.…
shuhalo
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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, 2) Via connections $1$-forms, 3) Via covariant…
Anthony Conway
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Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as a derivation. So now I read this apparent…
sifsa
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When is a connection on the adjoint bundle induced by a principal connection?

Let $P\rightarrow M$ be a principal $G$-bundle with a connection 1-form $\omega$. In a local trivialisation $\tau_U \colon U\rightarrow P_U$ ($U \subset M$) we can pull the connection back to the base manifold. $$\omega_U=\tau_U^*\omega=\omega^i…
balintm
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Covariant derivative of a vector field along a curve

In the following proposition of Do Carmo's Riemannian Geometry, I don't understand part (c) because $\frac{DV}{dt}$ is a vector field along the curve $c$, so it has domain $I$, while $\nabla_{\frac{dc}{dt}}Y$ is a vector field on $M$ with domain…
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