Riemannian manifold and general relativity

[ehrenfest]

Homework Statement

My book ("General Relativity" by Hobson on page 32) says that an N-dimensional manifold has 1/2 * N * (N-1) independent metric functions. I am confused about why there is a limit at all to the number of independent metric functions $$g_{\mu \nu}$$. It probably has to do with the symmetric property of the metric.

Homework Equations

The Attempt at a Solution

 

[Avodyne]

I think you're confused about what they mean by "indpendent functions". If we ignored symmetry considerations, the metric would have N² indepedent functions: g₁₁(x), g₁₂(x), ..., g_NN(x). But the metric must be symmetric, g_mn(x)=g_nm(x). The number of indepednet functions is then the number of elements on or above the main diagonal in an NxN matrix; this number is (1/2)N(N+1). [Not, as you wrote, (1/2)N(N-1); your book has the correct result.]

 

[ehrenfest]

Thank you. And you're right it is 1/2 * (N) * (N+1)