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:

1. On ${\mathbb{R}^n}$, there is a standard Riemannian metric coming from the usual inner product. More generally, if ${g_{ij}: \mathbb{R}^n \rightarrow \mathbb{R}}$ are smooth functions such that the matrix ${(g_{ij}(x))}$ is symmetric and positive definite for all ${x \in \mathbb{R}^n}$, we get a Riemannian metric ${\sum_{i,j} g_{ij} dx^i \otimes dx^j}$ on ${\mathbb{R}^n}$, where the sum is to be interpreted as a covariant tensor.
2. Given an immersion ${N \rightarrow M}$, a Riemannian metric on ${M}$ induces one on ${N}$ in the natural way, simply by pulling back. For instance, any surface in ${\mathbb{R}^3}$ has a Riemannian structure based upon the standard Riemannian structure on ${\mathbb{R}^3}$—based simply on the usual inner product–and induced on the surface.
3. Given an open covering ${U_i}$ on ${M}$, Riemannian metrics ${(\cdot, \cdot)_i}$ on ${U_i}$, and a partition of unity ${\phi_i}$ subordinate to the covering ${U_i}$, we get a Riemannian metric on ${M}$ by$\displaystyle (v,w)_p := \sum_i \phi_i(p) (v,w)_{i,p}.$Thus, using 1) above, any paracompact manifold–which necessarily admits partitions of unity–can be given a Riemannian metric.  (Edit: Paracompactness is automatic for smooth manifolds, as the standard definition of a smooth manifold apparently requires them to be second-countable.  This may vary though.)

A Riemannian metric allows us to take the length of a curve in a manner resembling the standard case. Given ${v \in T_p(M)}$, use the notation ${\left \lVert{v} \right \rVert := (v,v) = (v,v)_p}$. If ${c: I \rightarrow M}$ is a smooth curve for ${I}$ an interval in ${\mathbb{R}}$, we define

$\displaystyle l(c) := \int_I \left \lVert{c'(t)}\right \rVert dt;$

this is easily checked to be independent of parametrization, just as in the usual case. Using this, we can make a Riemannian manifold ${M}$ into a metric space: for ${p,q \in M}$, let

$\displaystyle d(p,q) := \inf_{c \mid c(a)=p,c(b)=q} l(c).$

Proposition 1 The metric on ${M}$ induces the standard topology on ${M}$.

This is a local question, so we can reduce to the case of ${M}$ an open ball in euclidean space ${\mathbb{R}^n}$. Each tangent vector ${v \in T_p(M)}$ can be viewed as an element of ${\mathbb{R}^n}$ in a natural way. Now let ${\left \lVert{\cdot}\right \rVert_{\mathbb{R}^n}}$ be the standard norm on ${\mathbb{R}^n}$. By continuity, we can find ${\delta>0}$ by shrinking ${M}$ if necessary such that for all ${v \in T_p(M), p \in K}$,

$\displaystyle \delta \left \lVert{v}\right \rVert_{\mathbb{R}^n} \leq \left \lVert{v}\right \rVert_p \leq \delta^{-1} \left \lVert{v}\right \rVert_{\mathbb{R}^n} ;$

in particular, the lengths of curves in ${M}$ are necessarily comparable to the usual lengths in ${\mathbb{R}^n}$. The result now follows. There are some more interesting questions that I will return to later–for instance, the question of when such a metric is complete. First, I will need to talk about connections.

Connections

In ${\mathbb{R}^n}$, as I have already alluded to above, there is a natural way of identifying any two tangent spaces with each other (and with ${\mathbb{R}^n}$); there is no such canonical way on an arbitrary manifold, but a connection gives us a good way of doing it. In particular, a connection gives a way to differentiate a vector field, which normally would not make sense outside of ${\mathbb{R}^n}$–this is how it is defined (according to Koszul, anyway). A connection on ${M}$ associates to vector fields ${X,Y}$ on ${M}$ another vector field ${\nabla_X Y}$ (called the covariant derivative of ${Y}$ with respect to ${X}$) satisfying the following conditions

1. ${\nabla}$ is linear in ${X}$ over the smooth functions on ${M}$, i.e. ${\nabla_{fX}(Y) = f \nabla_X Y}$.
2. ${\nabla}$ is a derivation in ${Y}$, i.e. for ${f: M \rightarrow \mathbb{R}}$ smooth,$\displaystyle \nabla_X(fY) = (Xf)Y + f \nabla_X Y.$

The standard example here to keep in mind is the case of ${M = \mathbb{R}^n}$, where given vector fields ${X = \sum X^i \frac{\partial}{\partial x_i}, Y = \sum Y_i \frac{\partial}{\partial x_i} }$, we set

$\displaystyle \nabla_X Y := \sum Y^i(X)\frac{\partial}{\partial x_i} \$

In fact, the definition is quite expansive, and allows us to construct many more examples. Given an open subset ${U \subset \mathbb{R}^n}$ and smooth functions (called Christoffel symbols) ${\Gamma_{ij}^k: U \rightarrow \mathbb{R}, 1 \leq i,j,k \leq n}$, we can construct a connection ${\nabla}$ on ${U}$ with ${\nabla_{\partial_i} \partial_j = \sum_k \Gamma_{ij}^k \partial_k}$, where ${\partial_i := \frac{\partial}{\partial x_i}}$. This is done simply by using 1 and 2 above.

Next I should probably talk about covariant derivatives along a curve, parallelism, the Levi-Civita connection associated to a Riemannian metric, geodesics, and completeness.  Also, eventually I’d like to discuss Ehresmann’s definition of a connection, but I’ll first have to talk about Lie groups (and understand his definition better).

Slightly corrected since the first post (notational changes, discussion of paracompactness, and an error in the definition of a connection).