One of the central tasks in differential geometry is to make precise the notion of length and angle on a smooth manifold. Unlike $\mathbb R^n$, a general manifold comes with no preferred inner product. The metric tensor is not something that “appears” automatically; rather, it is a geometric structure we deliberately introduce. The goal of this post is to derive the metric tensor from first principles in a clean, logically economical way.
We begin with a smooth $n$-dimensional manifold $M$. At each point $p\in M$, we want to measure lengths of infinitesimal displacements through $p$. Infinitesimal displacements are modeled by tangent vectors, so the correct object to define is an inner product on each tangent space $T_pM$.
A precise and coordinate-free definition of tangent vectors is the following. A tangent vector at $p$ is a derivation at $p$: a linear map $X:C^\infty(M)\to\mathbb R$ satisfying the Leibniz rule $X(fg)=f(p)X(g)+g(p)X(f)$. The collection of all such derivations forms a real vector space $T_pM$ of dimension $n$.
Now introduce a coordinate chart $(U,\varphi)$ with $\varphi=(x^1,\dots,x^n)$ and $p\in U$. For each coordinate function, define a derivation by
These derivations form a basis of $T_pM$. Every tangent vector $X\in T_pM$ can therefore be written uniquely as
for real coefficients $X^i$. These coefficients depend on the chosen coordinates, but the vector $X$ itself does not. At this point, nothing like length exists yet. To measure length, we must specify an inner product on each tangent space.
A Riemannian metric on $M$ is a rule that assigns to each point $p\in M$ a bilinear map
such that:
- $g_p$ is symmetric: $g_p(X,Y)=g_p(Y,X)$
- $g_p$ is positive definite: $g_p(X,X)>0$ for all $X\neq 0$
- The assignment $p\mapsto g_p(X_p,Y_p)$ is smooth whenever $X,Y$ are smooth vector fields
This definition is entirely coordinate-free. The metric is simply a smoothly varying inner product on tangent spaces.
Now fix a coordinate chart $(x^1,\dots,x^n)$. Since $g_p$ is bilinear, it is completely determined by its values on basis vectors. Define
These functions $g_{ij}$ are smooth, symmetric in $i,j$, and vary from point to point. They are the components of the metric tensor in coordinates.
Given two tangent vectors
bilinearity immediately gives
This is the familiar coordinate expression of the metric. Importantly, this is not a definition but a representation of the underlying geometric object $g$.
From this formula, the squared length of a tangent vector $X$ is
Thus, the metric tensor generalizes the Euclidean dot product by allowing the coefficients $g_{ij}$ to vary with position and coordinates.
It is instructive to check how these components transform. If we change coordinates from $x^i$ to $\tilde x^a$, then the basis vectors transform via the chain rule:
Substituting into the definition of the metric components yields
This transformation law is exactly what characterizes $g_{ij}$ as the components of a $(0,2)$-tensor field. The metric tensor is therefore not merely a matrix-valued function but a genuine tensorial object on the manifold.