For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.
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 $$H^2=\left(\mathbb{R}\times\mathbb{R}_+,\frac{1}{y^2}\left(\mathrm{d}x\otimes\mathrm{d}x+\mathrm{d}y\otimes\mathrm{d}y\right)\right).$$
This tag is for questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag if your question involves Riemannian manifolds or objects generally associated with them, such as Levi-Civita connections.
