I wonder if there are standard extensions of the concept of solid angle in curved space.
There should be because physicists who study radiation in curved space would run into this kind of needs.
If there exist such definitions, where can I find web-references on the topic ?
Thanks beforehand !
The notion of solid angle exists and is infinitesimal: Let $M$ be a Riemannian manifold and $S\subset M$ be an open subset whose closure is, say, locally convex. (You need some regularity assumptions on $S$ to make sense of the definition below.) Suppose that $x$ is a boundary point of $S$. Define the tangent cone $C_x(S)$ of $S$ at $x$ as the set of velocity vectors $p'_+(0)\in T_x(M)$, for all paths $p: [0, \infty)\to S$, $p(0)=x$, such that the derivative from the right, $p'_+(0)$, exists. By the convexity assumption, $C_x(S)$ is a convex cone (it is stable under multiplication by positive real numbers).
Example: If $S$ has smooth boundary at $x$ then $C_x(S)$ is a closed half-space in $T_xM$.
Let $\Omega$ denote the intersection $C_x(S)\cap T^1_x(M)$, where $T^1_x(M)$ is the unit sphere in $T_x(M)$ centered at the origin. Now, the solid angle of $S$ at $x$ is just the total area of $\Omega$: If $M$ is $n$-dimensional, then this area will be the $n-1$-dimensional measure of $C_x(S)\cap T^1_x(M)$.