Riemannian Geometry
Level: 0 1 2 3 4 5 6 7
MSC Classification: 53 (Differential geometry)

Prerequisites: Curves and Surfaces

Getting Oriented

In general, geometry is the study of spaces which have some notion of distance. Differentiable geometry adds such a notion to topological spaces by requiring the spaces to locally "look like" $\mathbb{R}^n$. One can then analyze the space (called a manifold) by extending results on $\mathbb{R}^n$ to the manifold. This becomes especially fruitful if the manifold is given a \emph{Riemannian metric}, which intuitively speaking is a notion of distance. This allows ‘calculus’ on the manifold, and forms the basis for \emph{Riemannian geometry}.

The most fundamental theorems of geometry are, perhaps, the Inverse Function Theorem and its twin the Implicit Function Theorem……

Beyond the basics of manifolds, one delves into the consequences of a Riemannian structure. One requires this structure because abstract manifolds are not embedded in $\mathbb{R}^n$, and therefore do not have an intrinsic notion of distance……


The Basics

First we will consider manifolds without the Riemannian structure. The formal definition defines a manifold as several ‘pieces’ of $\mathbb{R}^n$ glued together in a suitable fashion. This is analogous to mapping the earth with an atlas of maps, each covering a small piece (hence the terms map, chart, and atlas).

(of dimension $n$) a set $M$ which is the union of open sets $V_\alpha$ homeomorphic to $U_\alpha \subset \mathbb{R}^n$ via maps $x_\alpha: U_\alpha \to V_\alpha$ such that if $V_\alpha \cap V_\beta \neq \emptyset$ then the transition function $x_\beta^{-1} \circ x_\alpha$ is differentiable (as a map from $\mathbb{R}^n \to \mathbb{R}^n$). The pairs $(U_\alpha,x_\alpha)$ are called coordinate charts and the family $\{(U_\alpha,x_\alpha)\}$ a differentiable structure (or atlas).

The simplest examples of manifolds are the spaces $\mathbb{R}^n$, and the spheres $S^n$. Clearly, $\mathbb{R}^n$ requires only one coordinate chart, while $S^n$ requires two, each of which covers all but one point of the sphere (use stereographic projection).

Properties of differentiability in $\mathbb{R}^n$ are passed on to manifolds, by considering the local coordinates. Thus, a map $f: M \to N$ is differentiable if each map $y_\beta\circ f \circ x_\alpha^{-1}$ is differentiable.

The Implicit and Inverse Function Theorems

The closely-related implicit and inverse function theorems are the foundation of differential geometry. The proof uses a little analysis, briefly described here. First, a contraction on a metric space $X$ is a map $T: X \to X$ together with a constant $K>0$ such that $d(Tx,Ty)\leq K d(x,y)$ always holds. The Banach Contraction Principle states that any contraction has a unique fixed point $\xi=\lim T^n(X)$.

Applying this result to the space of functions with the uniform metric, one obtains:

Implicit Function Theorem
Given a continuous map $g: \mathbb{R}^n\times\mathbb{R}^m \to \mathbb{R}^m$ defined on a neighborhood of $(\xi,\eta)$ with $g(\xi,\eta)=0$ such that the differential of the map $y \mapsto g(\xi,y)$ is surjective at $\eta$ (so the Jacobian $J(g_i,y_j)$ is nonzero), there exists a unique continuous map $\phi$ taking an open ball around $\xi$ to an open ball around $\eta$, with $\phi(\xi)=\eta$ and $g(x,\phi(x))=0$.

This may sound like a mouthful, but all it means is that the implicit relation $g(x,y)=0$ can be solved locally for $y$ in terms of $x$ whenever $g$ depends ‘nicely’ on $y$. The proof applies the Banach contraction principle to the space of functions with the uniform metric.

A corollary of the implicit function theorem is:

Inverse Function Theorem
Given a continuous map $f: \mathbb{R}^m \to \mathbb{R}^m$ with nonsingular differential at $\xi$ and $f(\xi)=\eta$, there exists a unique continuous map $\phi$ between open balls around $\xi$ and $\eta$ with $\phi(\eta)=\xi$ and $f\circ\phi=\mathsf{Id}$.

So, every ‘nice’ function $f(x)$ has a local inverse. The proof merely applies the implicit function theorem to $g(x,y)=f(y)-x$.

More on Manifolds…

The Riemannian Metric


The Road Ahead

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