**18**

votes

**3**answers

3k views

### Geometrical meaning of the Ricci Tensor and its Symmetry

Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume is that it has zero ...

**5**

votes

**1**answer

680 views

### Symmetric Ricci Tensor

Let $M$ be a pseudo-Riemannian manifold. Assume that $M$ comes with a zero-torsion affine connexion $\nabla.$ There is no need for $\nabla$ to be the Levi-Civita connexion. Recall that the curvature ...

**6**

votes

**1**answer

588 views

### Almost Flat Connections On Principal G-Bundles

Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$.
We know ...

**12**

votes

**2**answers

1k views

### The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...

**1**

vote

**2**answers

1k views

### tangent and cotangent bundle

Hi, I am reading "Introduction to symplectic topology" by McDuff and salamon. At some point I cant go further. My question is: Let $(M,g)$ be a Riemannian manifold and consider the cotangent bundle $T^...

**6**

votes

**3**answers

905 views

### For quasi-coherent D-Modules

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group ...

**4**

votes

**1**answer

413 views

### “Nash Style” Embedding Theorem for Connections

The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian ...

**11**

votes

**3**answers

4k views

### Why is the integral of the second chern class an integer?

I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon.
Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is ...

**3**

votes

**1**answer

638 views

### Holonomy Groups and the Hopf Fibration

I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a ...

**5**

votes

**1**answer

639 views

### Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...

**1**

vote

**0**answers

266 views

### Non-Existence of a Principal Connection for the Sphere over Projective Space?

As of the Wikipedia article on principal bundles connections:
Let $\pi: P \to M$ be a smooth principal bundle, a principal $G$-bundle over a smooth manifold $M$. Then a principal $G$-connection on ...

**20**

votes

**4**answers

2k views

### Rolling without slipping interpretation of torsion

This is, in a sense, a follow up to this question.
Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...

**2**

votes

**0**answers

184 views

### Hermitian connections on real hypersurfaces of $\mathbb C^{n+1}$

I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real ...

**3**

votes

**2**answers

995 views

### Why must a reducible flat SU(2)-connection over a homology sphere be trivial?

Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation ...

**12**

votes

**3**answers

510 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 \...

**11**

votes

**5**answers

3k 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

**0**answers

473 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

**1**answer

418 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

**5**answers

4k 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 Levi-...

**4**

votes

**4**answers

955 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 ...

**6**

votes

**1**answer

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

**0**answers

635 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 ...

**8**

votes

**1**answer

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

**1**answer

225 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

**1**answer

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 ...