A Riemannian metric tensor is a covariant, type $(0,2)$ tensor field $g\in\Gamma(\rT^* M\otimes \rT^*M)$ such that at each point $p\in M$ , the bilinear form $g_p:\rT_pM\times \rT_p M\to \Rset$ is symmetric and positive definite. Here $T^* M$ is the cotangent bundle of $M$ (defined as a sheaf), $\Gamma$ is the set of global sections of $T^* M \otimes T^* M$ , and $g_p$ is the value of the function $g$ at the point $p \in M$ .

Let $(x^1,\ldots,x^n)$ be a system of local coordinates on an open subset $U\subset M$ , let $dx^i,\; i=1,\ldots, n$ be the corresponding coframe of 1-forms, and let $\displaystyle \ddx{i},\; i=1,\ldots, n$ be the corresponding dual frame of vector fields. Using the local coordinates, the metric tensor has the unique expression $$g=\sum_{i,j=1}^n g_{ij}\, dx^i\otimes dx^j,$$ where the metric tensor components $$g_{ij}=g\left(\ddx{i},\ddx{j}\right)$$ are smooth functions on $U$ .

Once we fix the local coordinates, the functions $g_{ij}$ completely determine the Riemannian metric. Thus, at each point $p\in U$ , the matrix $(g_{ij}(p))$ is symmetric, and positive definite. Indeed, it is possible to define a Riemannian structure on a manifold $M$ by specifying an atlas over $M$ together with a matrix of functions $g_{ij}$ on each coordinate chart which are symmetric and positive definite, with the proviso that the $g_{ij}$ 's must be compatible with each other on overlaps.

A manifold $M$ together with a Riemannian metric tensor $g$ is called a Riemannian manifold.

Note: A Riemannian metric tensor on $M$ is not a distance metric on $M$ . However, on a connected manifold every Riemannian metric tensor on $M$ induces a distance metric on $M$ , given by $$ d(x,y) := \inf \left\{ \int_0^1 \left[g\!\!\left( \frac{dc}{dt}, \frac{dc}{dt}\right)_{\!c(t)}\right]^{1/2}dt \right\} ,\quad x,y\in M,$$ where the infimum is taken over all rectifiable curves $c:[0,1]\to M$ with $c(0)=x$ and $c(1)=y$ .

Often, it is the $g_{ij}$ that are referred to as the ``Riemannian metric''. This, however, is a misnomer. Properly speaking, the $g_{ij}$ should be called local coordinate components of a metric tensor, where as ``Riemannian metric'' should refer to the distance function defined above. However, the practice of calling the collection of $g_{ij}$ 's by the misnomer ``Riemannian metric'' appears to have stuck.

Remarks:

• Both the Riemannian manifold and Riemannian metric tensor are fundamental concepts in Einstein's General Relativity (GR) theory where the ``Riemannian metric'' and curvature of the physical Riemannian space-time are changed by the presence of massive bodies and energy according to Einstein's fundamental GR field equations.
• The category of Riemannian manifolds (or `spaces') provides an alternative framework for GR theories as well as algebraic quantum field theories (AQFTs);
• The category of `pseudo-Riemannian' manifolds, deals in fact with extensions of Minkowski spaces, does not possess the Riemannian metric defined in this entry on Riemannian manifolds, and is claimed as a useful approach to defining 4D-spacetimes in relativity theories.