Part 1
Chapter 1 Definition: Riemannian Metric
A Riemannian metric on a smooth manifold $M$ is a family of inner products $g_p : T_pM \times T_pM \to \mathbb{R}$, one for each point $p \in M$, that varies smoothly from point to point. That is, $g_p$ is bilinear, symmetric, and positive-definite. By smoothly varying, we mean that for any two smooth vector fields $X$ and $Y$ on $M$, the function $p \mapsto g_p(X_p, Y_p)$ is a smooth function.
The pair $(M, g)$ is called a Riemannian manifold.
Local Coordinates
In a local coordinate chart $(U, x^i)$, the metric is expressed as a symmetric $(0,2)$-tensor:
\[g = g_{ij} dx^i \otimes dx^j\]The components $g_{ij} = g(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}) $ form a symmetric, positive-definite matrix at each point. This matrix allows us to compute the inner product of two tangent vectors $v = v^i \frac{\partial}{\partial x^i}$ and $w = w^j \frac{\partial}{\partial x^j}$ as:
\[g(v, w) = g_{ij}v^i w^j\]Note: The Einstein summation convention is used here. Repeated indices, usually one upper and one lower, are summed from $1$ to $n$, the dimension of $M$. For example, $g_{ij} dx^i \otimes dx^j = \sum_{i,j=1}^n g_{ij} dx^i \otimes dx^j$.
Example: The Euclidean Metric
The simplest example of a Riemannian metric is the standard Euclidean metric on $\mathbb{R}^n$. In the standard coordinates $(x^1, \dots, x^n)$, the metric tensor is given by the Kronecker delta, $\delta_{ij}$.
The matrix of components is the identity matrix:
\[[g_{ij}] = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}\]The Lean definition of Riemannian metric.