## Product metric on product manifold

Suppose we have two Riemannian manifolds and , what happened to their product manifold with product metric?

.

We will use and . Then by definition . Therefore

Similarly for all mixed indices on . This implies

Therefore the Riemannian curvature operator is

and Ricci tensor

When you have the warped product metric like with metric , the curvature is given like the following

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