@Igor Belegradek Thank you, I will try to read the book.

But I'm still confused that the $\delta$ in locally injective domain can be made uniformly for all $s\in S$, though $S$ is compact.

Thank you for your explanation.

I think this can be understand as the index theorem for focal points?

Thank you for your excellent answer. And I have one more question: it seems that we just need these two metrics concide at the origin up to the first order because to calculate the principal curvature we only need the Christoffel symbols.