The Metric Tensor Of A Cartesian Product

Question

Hello I'm a physics student, not a math student, who is trying to understand how to write down the metric of a manifold which is constructed via a cartesian product of two other manifolds I know the metrics for. For example S^1 X S^1 is a Torus and I know the metric for a Circle, and I even know the metric for a Torus, but I do not know how to derive the metric for a Torus from the Cartesian product of two circles. In the end I want to construct the metric for the S^1 X S^2. Any help will be appreciated thank you.

Answer 1

Given two Riemannian Manifolds $(M,g)$ and $(N,h)$ then we define their product metric at the point $(p,q)\in M\times N$ as $$g \oplus h: (T_pM\times T_qN) \times (T_pM\times T_qN) \rightarrow \mathbb R$$ $$(g\oplus h)_{(p,q)}((v,w),(\hat v,\hat w)):=g_p(v,\hat v)+h_q(w,\hat w)$$