Calculating the area and length of sets using a Riemannian metric on the sphere

Question

Let $S^2\subseteq \mathbb{R}^3$ be the unit sphere. Let's define the Riemannian metric to be $d(x,y)=\angle(x,y)=\arccos(x,y)$. Calculate the area and circumference of the ball $B(x,R)=\left\{y\in S^2: d(x,y)\leq R\right\}$ . So I'm not quite sure how to approach this question without writing the $g$ matrix for this metric, and I'm not quite sure how to write this matrix. I also thought of doing a coordinate change to spherical coordinates, but I'm still not quite sure how to write the matrix under these coordinates. Thanks a lot for the help.I'd also appreciate a good online book about differential geometry.

Answer 1

Terminology note: Your "d" is the topological metric (i.e., distance function) associated to the round metric on the unit sphere. (The latter is the "Riemannian" metric, a bilinear function on tangent vectors.)

The round metric is the restriction of the Euclidean metric on $\mathbf{R}^3$ to the unit sphere $S^2$. If you parametrize $S^2$ by (your favorite version of) spherical coordinates, denoting the mapping $\Phi(u, v)$ and using subscripts to denote partial derivatives, then the round metric in your coordinate chart has component functions $$ g_{11} = \Phi_u \cdot \Phi_u,\quad g_{12} = g_{21} = \Phi_u \cdot \Phi_v,\quad g_{22} = \Phi_v \cdot \Phi_v. $$ (Classically these functions are usually denoted $E$, $F$,and $G$.) The area element is $$ dA = \sqrt{\det g}\, du\, dv = \sqrt{EG - F^2}\, du\, dv, $$ and the area integral is straightforward to evaluate.