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…)