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Introduction

Metaphorically, Riemannian geometry is what happens when we try to generalize the Pythagorean theorem to work on smooth manifolds in general, but accidently drop the Pythagorean theorem in a blender along the way.

Definition 1.1a: (Riemannian Metric) Suppose $M$ is a smooth manifold. A Riemannian metric on $M$ is a section $\mathrm{g}\in\Gamma(T^\vee\hspace{-.25em}M\otimes T^\vee\hspace{-.25em}M)$ such that, for each $p\in M$ and all $X_p,Y_p\in T_pM$,

  • $\mathrm{g}_p(X_p\otimes Y_p)=\mathrm{g}_p(Y_p\otimes X_p)$,

  • $\mathrm{g}_p(X_p\otimes X_p)\geq0$, with equality if and only if $X_p=0$.

Note that many mathematicians use the following equivalent definition.

Definition 1.1b: (Riemannian Metric) Suppose $M$ is a smooth manifold. A Riemannian metric is a smooth function $\mathrm{g}:TM\times_MTM\to\mathbb{R}$, where $TM\times_MTM$ is the fiber product, such that, for each $p\in M$, all $X_p,Y_p,Z_p\in T_pM$, and all $a,b\in\mathbb{R}$,

  • $\mathrm{g}(aX_p+bY_p,Z_p)=a\mathrm{g}(X_p,Z_p)+b\mathrm{g}(Y_p,Z_p)$,
  • $\mathrm{g}(X_p,Y_p)=\mathrm{g}(Y_p,X_p)$,
  • $\mathrm{g}(X_p,X_p)\geq0$, with equality if and only if $X_p=0$.

Regardless of the particulars of the definition, a Riemannian metric is essentially a smooth choice of inner product on each tangent space. Making a choice of Riemannian metric gives us a Riemannian manifold.

Definition 1.2: (Riemannian Manifold) A Riemannian manifold is a pair $(M,\mathrm{g})$, where $M$ is a smooth manifold and $\mathrm{g}$ is a Riemannian metric.

There is a plethora of examples of Riemannian manifolds that appear all over geometry.

Example 1.3: (Euclidean Space) Let $x$ be the (global) identity chart on $\mathbb{R}^n$. A Euclidean space is a Riemannian manifold of the form $$\left(\mathbb{R}^n,\sum_{i=1}^n\mathrm{d}x^i\otimes\mathrm{d}x^i\right).$$ Usually, we identify $n$-dimensional Euclidean space with $\mathbb{R}^n$.

Example 1.4: (Hyperbolic Plane) Let $(x,y)$ be the (global) identity chart on the upper half plane. Then, the hyperbolic plane is the Riemannian manifold $$H^2=\left(\mathbb{R}\times\mathbb{R}_+,\frac{1}{y^2}\left(\mathrm{d}x\otimes\mathrm{d}x+\mathrm{d}y\otimes\mathrm{d}y\right)\right).$$

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