Defining Metrics on Submanifolds: Is G = goi* a Valid Approach?

[math6]

can we define a metric on a submanifold of as follows:
M is a manifold equipped with a Riemannian metric g, we denote (M, g), and S is a submanifold of M.
an application i of S to M is an immersion, i * is the linear tangent; then a metric on S G is given by G = goi*.

 

[arkajad]

Provided that you understand that i* acts as

$$G(\xi,\eta)=g(i^*\xi,i^*\eta)$$

If $$(q^i)=(q1,\ldots ,q_p)$$ are parameters on the submanifold, and $$x^\mu=(x_1,\ldosts ,x_n)$$ are (local) coordinates on the manifold, then, in coordinates, this becomes

$$g_{ij}=\frac{\partial x^\mu}{\partial q^i}\frac{\partial x^\nu}{\partial q^j}\,g_{\mu\nu}$$

 

[math6]

thnx but i just doubt about something and i just want to be confirmed .