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12
votes
3answers
500 views

Can a homotopy inverse of the map from a Lie group to loops on its classifying space be given by holonomy?

Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds: $$BG \simeq ...
9
votes
5answers
2k views

pull-back connection

I have a question related to the definition of the pull-back connection, more specifically about its uniqueness or the canonical way to induce it. The definition that one finds in general goes ...
5
votes
0answers
464 views

Flat connections with logarithmic poles and p-adic parallel transport under choice of branch of logarithm

Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically ...
1
vote
1answer
411 views

notion of connection on Torsors

Hi, Can anyone please enlighten me on the notion of connection for G-torsors? Edit (by WW) For some reason the OP decides to add clarification below as an answer rather than editing the question. ...
26
votes
5answers
3k views

A geometric interpretation of the Levi-Civita connection?

Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the ...
4
votes
4answers
943 views

Proving the basic identity which implies the Chern-Weil theorem

If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $\Omega^p(M,E) = \Gamma ( \wedge ^p (T^*M) \otimes E) $ The ...
5
votes
1answer
2k views

Skellam distribution: Deep connection between Poisson distributions and Bessel function?

The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by: $$ f(k;\mu_1,\mu_2)= ...
3
votes
0answers
621 views

Gauss--Manin connection for de Rham realisation

Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a ...
7
votes
1answer
1k views

Global description of the Levi-Civita connection

I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X. I'm not looking for a description of this ...
2
votes
1answer
222 views

Immersion with respect to a connection?

In the paper hep-th/9712042v2, p. 20, the following setup is given: A complex manifold M and an n+1-dimensional vector bundle V on it. V has an underlying real bundle $V_{\mathbb{R}}$ with a flat ...
10
votes
1answer
1k views

Frobenius Descent

Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...