Editing Riemannian manifold (section)

=== Riemannian metrics and Riemannian manifolds ===
[[File:Tangent plane to sphere with vectors.svg|thumb|A tangent plane of the sphere with two vectors in it. A Riemannian metric allows one to take the inner product of these vectors.]]

Let <math>M</math> be a [[smooth manifold]]. For each point <math>p \in M</math>, there is an associated vector space <math>T_pM</math> called the [[tangent space]] of <math>M</math> at <math>p</math>. Vectors in <math>T_pM</math> are thought of as the vectors tangent to <math>M</math> at <math>p</math>.

However, <math>T_pM</math> does not come equipped with an [[inner product]], a measuring stick that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts a measuring stick on every tangent space. 

A ''Riemannian metric'' <math>g</math> on <math>M</math> assigns to each <math>p</math> a [[positive-definite]] inner product <math>g_p : T_pM \times  T_pM \to \mathbb R</math> in a smooth way (see the section on regularity below).{{sfn|do Carmo|1992|p=38}} This induces a norm <math> \|\cdot\|_p : T_pM \to \mathbb R</math> defined by <math>\|v\|_p = \sqrt{g_p(v,v)}</math>. A smooth manifold <math>M</math> endowed with a Riemannian metric <math>g</math> is a ''Riemannian manifold'', denoted <math>(M,g)</math>.{{sfn|do Carmo|1992|p=38}} A Riemannian metric is a special case of a [[metric tensor]].

A Riemannian metric is not to be confused with the distance function of a [[metric space]], which is also called a metric.

====The Riemannian metric in coordinates====
If <math>(x^1,\ldots,x^n):U\to\mathbb{R}^n</math> are smooth [[local coordinates]] on <math>M</math>, the vectors
: <math>\left\{\frac{\partial}{\partial x^1}\Big|_p,\dotsc, \frac{\partial}{\partial x^n}\Big|_p\right\}</math>
form a basis of the vector space <math>T_pM</math> for any <math>p\in U</math>. Relative to this basis, one can define the Riemannian metric's components at each point <math>p</math> by
: <math>g_{ij}|_p:=g_p\left(\left.\frac{\partial }{\partial x^i}\right|_p,\left.\frac{\partial }{\partial x^j}\right|_p\right)</math>.{{sfn|Lee|2018|p=13}}
These <math>n^2</math> functions <math>g_{ij}:U\to\mathbb{R}</math> can be put together into an <math>n\times n</math> matrix-valued function on <math>U</math>. The requirement that <math>g_p</math> is a positive-definite inner product then says exactly that this matrix-valued function is a [[symmetric matrix|symmetric]] [[positive-definite matrix|positive-definite]] matrix at <math>p</math>.

In terms of the [[tensor algebra]], the Riemannian metric can be written in terms of the [[dual basis]] <math>\{ dx^1, \ldots, dx^n \}</math> of the cotangent bundle as
: <math> g=\sum_{i,j}g_{ij} \, dx^i \otimes dx^j.</math>{{sfn|Lee|2018|p=13}}

==== Regularity of the Riemannian metric ====
The Riemannian metric <math>g</math> is ''continuous'' if its components <math>g_{ij}:U\to\mathbb{R}</math> are continuous in any smooth coordinate chart <math>(U,x).</math> The Riemannian metric <math>g</math> is ''smooth'' if its components <math>g_{ij}</math> are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as [[Lipschitz continuity|Lipschitz]] Riemannian metrics or [[Measurable function|measurable]] Riemannian metrics.

There are situations in [[geometric analysis]] in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). However, in this article, <math>g</math> is assumed to be smooth unless stated otherwise.

==== Musical isomorphism ====
{{Main|Musical isomorphism}}

In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its [[dual vector space|dual]] given by <math>v \mapsto \langle v, \cdot \rangle</math>, a Riemannian metric induces an isomorphism of bundles between the [[tangent bundle]] and the [[cotangent bundle]]. Namely, if <math>g</math> is a Riemannian metric, then
: <math>(p,v) \mapsto g_p(v,\cdot)</math>
is a isomorphism of [[smooth vector bundle]]s from the tangent bundle <math>TM</math> to the cotangent bundle <math>T^*M</math>.{{sfn|Lee|2018|p=26}}