(1) If $N^k$ is a submanifold in a compact Riemannian manifold $M^{k+m},\ m\geq 1$ s.t. each $p\in N$ has the following property : There exists independent set $\{ X_i\}_{i=1}^k$ tangent to $T_pN$ s.t. $$\sum_{i=1}^k( \nabla_{X_i}n,{X_i})=0$$ for any $n$ which is any unit normal to $N$, then $N$ is not totally geodesic in general. Is this true ?

If so, can you give an example ?

In $M={\bf R}^3$, i.e., $M$ is noncompact, hyperbolid is not totally geodesic but it satisfies the above condition.

(2) We can give an example under more strong condition ? : $ (\nabla_{X_i}n,{X_i})=0$ for all $i$

Thank you in anticipaction

traceof the Weingarten map $X\mapsto \nabla_X n$ vanishes, so a minimal surface will not satisfy $\langle\nabla_Xn,X\rangle=0$ for every $X$. If I understand correctly your condition is the vanishing of the Weingarten map (or second fundamental form for that matter form), which is equivalent to being totally geodesic. $\endgroup$ – Thomas Richard Mar 17 '16 at 9:53