Editing Riemannian manifold (section)

====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}}