Recall that two Riemannian manifolds ${M,N}$ are isometric if there exists a diffeomorphism ${f: M \rightarrow N}$ that preserves the metric on the tangent spaces. The curvature tensor  (associated to the Levi-Civita connection) measures the deviation from flatness, where a manifold is flat if it is locally isometric to a neighborhood of ${\mathbb{R}^n}$.

Theorem 1 (The Test Case) The Riemannian manifold ${M}$ is locally isometric to ${\mathbb{R}^n}$ if and only if the curvature tensor vanishes. (more…)

It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric ${g}$ has to satisfy certain conditions.  (As usual, we denote by $\nabla$ the Levi-Civita connection associated to $g$, and we assume the ground manifold is smooth.)

First of all, we have skew-symmetry

$\displaystyle R(X,Y)Z = -R(Y,X)Z.$

This is immediate from the definition.

Next, we have another variant of skew-symmetry:

Proposition 1 $\displaystyle g( R(X,Y) Z, W) = -g( R(X,Y) W, Z)$  (more…)

Ok, now onto the Levi-Civita connection. Fix a manifold ${M}$ with the pseudo-metric ${g}$. This means essentially a metric, except that ${g}$ as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.

This is the fundamental theorem of Riemannian geometry:

Theorem 1 There is a unique symmetric connection ${\nabla}$ on ${M}$ compatible with ${g}$. (more…)

Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold ${M}$ with a connection ${\nabla}$. Then parallel translation along a curve ${c}$ beginning at ${p}$ and ending at ${q}$ leads to an isomorphism ${\tau_{pq}: T_p(M) \rightarrow T_q(M)}$, which depends smoothly on ${p,q}$. For any ${r,s}$, we get isomorphisms ${\tau^{r,s}_{pq} :T_p(M)^{\otimes r} \otimes T_p(M)^{\vee \otimes s} \rightarrow T_q(M)^{\otimes r} \otimes T_q(M)^{\vee \otimes s} }$ depending smoothly on ${p,q}$. (Of course, given an isomorphism ${f: M \rightarrow N}$ of vector spaces, there is an isomorphism ${M^* \rightarrow N^*}$ sending ${g \rightarrow g \circ f^{-1}}$—the important thing is the inverse.) (more…)

Wow.  Blogging is definitely way harder during the academic year.

Ok, so I’m aiming to change things around a bit here and take a break from algebraic number theory to do some differential geometry. I’ll assume basic familiarity with what manifolds are, the tangent bundle and its variants, but generally no more. I eventually want to get to some real theorems, but this post will focus primarily on definitions.

Riemannian Metrics

A Riemannian metric on a smooth manifold ${M}$ is defined as a covariant symmetric 2-tensor ${(\cdot, \cdot)_p, p \in M}$ such that ${(v,v)_p > 0}$ for all ${v \in T_p(M)}$. For convenience, I will write ${(v,w)}$ for ${(v,w)_p}$. In other words, a Riemannian metric is a collection of (positive) inner products on each of the tangent spaces ${T_p(M)}$ such that if ${X,Y}$ are (smooth) vector fields, the function ${(X,Y): M \rightarrow \mathbb{R}}$ defined by taking the inner product at each point, is smooth. There are several ways to get Riemannian metrics: (more…)