Question on example of a Riemannian metric on $\mathbb{R}^n$

Question

Could anyone give me a non-trivial example (not the usual inner product) of a Riemannian metric on $\mathbb{R}^n$, considering $\mathbb{R}^n$ as a diffrentiable manifold of dimension $n$?

As I knew, a Riemannian metric is a collection of inner products, each of which is defined on tangent space of a point on the manifold (for specific, $\mathbb{R}^n$). However, I do not know what a tangent space of a point in $\mathbb{R}^n$ would be.

Thank you very much.

Answer 1

As a rather trivial example take some nonconstant function $\phi:\mathbb{R}^n \rightarrow \mathbb{R}^+$ and let $$g(x)(v,w) := \phi(x) \langle v, w\rangle$$ (You can think of the tangent space $T_x \mathbb{R}^n$ to $\mathbb{R}^n$ as a copy of $\mathbb{R}^n$ attached to the point $x$).

More generally, each smooth map $\psi:\mathbb{R}^n \rightarrow Mat(n)^+$ of $\mathbb{R}^n $ into the set of positive definite matrices defines a metric by $$g(v,w): = v^t \psi(x)w$$ (with $v,w\in T_x \mathbb{R}^n$)

Answer 2

Let be the following metric given by the following blockmatrix $$g_{ij}=\left(\begin{array}{ccc} 1 & 0 & \sin\vartheta_1\\ 0 & I & 0\\ \sin\vartheta_1 & 0 & 1 \end{array}\right),$$ where $I$ is the identity matrix. This is a nice metric that works out nicely in many calculations.