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Riemannian manifold
A Riemannian metric tensor is a covariant, type $(0,2)$ tensor field $g\in\Gamma(\operatorname{T}^{*}M\otimes\operatorname{T}^{*}M)$ such that at each point $p\in M$, the bilinear form $g_{p}:\operatorname{T}_{p}M\times\operatorname{T}_{p}M\to\mathbb{R}$ 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 1forms, and let $\displaystyle\frac{\partial}{\partial x^{{i}}},\;i=1,\ldots,n$ 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
$g_{{ij}}=g\left(\frac{\partial}{\partial x^{{i}}},\frac{\partial}{\partial x^{% {j}}}\right)$ 
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 spacetime 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 ‘pseudoRiemannian’ 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 4Dspacetimes in relativity theories.
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