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

Prerequisites: Curves and Surfaces

# Getting Oriented

 Rough Guides to Topology
 General Topology Algebraic Topology Manifold Theory Knot Theory

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……

# Manifolds

## 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).

Manifold
(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 $g(x,y)=f(y)-x$.