A Riemannian metric on a Differentiable manifold $M$ is a section $g$ of $T^* X\otimes T^* X$ (that is the Vector bundle of Tensor product of Cotangent spaces) which at each point is symmetric and Positive definite.

See here

If we have a Smooth function $F: M \to N$ between two manifolds, and a metric $g$ on $N$, we can pull back $g$ to a metric on $M$ $F^* g$. $F^* g$ is, by definition, a section of $T^*M \otimes T^* M$, which can be defined by specifying the element of $T^*_x \otimes T^*_x$ at each $x$. These elements of the tensor product of Cotangent spaces are by definition bilinear functions on Tangent vectors, so we can specify it by saying how it acts on two tangeng vectors $X$ and $Y$. We will define this action by mapping these tangent vectors to $N$ via the Differential of $F$ at $x$ ($DF_x$), and then using the matrix in $N$, at the corresponding point $F(x)$:

$(F^* g)_x (X,Y) = g_{F(x)} (DF_x(X), DF_x(Y))$.

A Diffeomorphism which preserves metric, under pullback, is called an Isometry.

A Riemannian metric defines a Length of Tangent vectors, as well as their Angles