I'm not completely sure what OP is really asking but here are some comments to the question (v3):

1. If $N\subseteq M$ is a $n$-dimensional submanifold inside a $m$-dimensional Riemanian manifold $(M,g)$, $n\leq m$, then $N$ has an induced metric $h\in\Gamma(T^*\!N\otimes T^*\!N)$ from the ambient space $(M,g)$.

2. If furthermore the tangent space $TN \subseteq TM$ is generated by vector fields $X_{(1)}, \ldots, X_{(n)}$, then the induced metric in a point $p\in N$ is given as $$ h_p (X_{(i)p},X_{(j)p})~=~g_p (X_{(i)p},X_{(j)p}), \qquad 1~\leq ~i,j~ \leq~ n . $$

I'm not completely sure what OP is really asking but here are some comments to the question (v3):

1. If $N\subseteq M$ is a $n$-dimensional submanifold inside a $m$-dimensional Riemannian manifold $(M,g)$, $n\leq m$, then $N$ has an induced metric $h\in\Gamma(T^*\!N\otimes T^*\!N)$ from the ambient space $(M,g)$.

2. If furthermore the tangent space $TN \subseteq TM$ is generated by vector fields $X_{(1)}, \ldots, X_{(n)}$, then the induced metric in a point $p\in N$ is given as $$ h_p (X_{(i)p},X_{(j)p})~=~g_p (X_{(i)p},X_{(j)p}), \qquad 1~\leq ~i,j~ \leq~ n . $$