**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-...

**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 ...

**8**

votes

**3**answers

747 views

### Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-...

**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 ...

**5**

votes

**1**answer

678 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 ...

**3**

votes

**1**answer

477 views

### The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...

**3**

votes

**2**answers

214 views

### Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.
Let $X$ be a ...

**4**

votes

**2**answers

405 views

### Chern class of a logarithmic connection

Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c_1(E)$ ...

**1**

vote

**0**answers

268 views

### The “Rolle theorem” for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...