(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 



Some manifold which is not totally geodesic in a compact manifold
Let us consider the case of hypersufaces for the sake of simplicity. The higher codimension case follows by taking products with $(S^1)^k$. 



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> *Observation :* Minimal submanifolds satisfy both of your conditions. In fact in this case the vectors $X\_i$ can be taken orthonormal. In fact if you ask for $X\_i$ to be orthonormal your first condition asks for the second fundamental form $B(X,Y)=\langle \nabla\_Xn,Y\rangle$ to have zero trace (so the mean curvature is minimal), while the second condition asks for the existence of an orthonormal basis of vectors which are isotropic for $B$ which is equivalent to the vanishing of the trace for a quadratic form. 
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So if we can find a compact manifold with a non totally geodesic minimal hypersurface, we can answer yes to both of your questions. Schwarz's $P$-surface in $\mathbb{R}^3$ is invariant by the action of $\mathbb{Z}^3$ by translation, and is not totally geodesic, so that it gives a non totally geodesic minimal surface in $\mathbb{T}^3$.


See here for Schwartz surface : [wiki](https://en.wikipedia.org/wiki/Schwarz_minimal_surface).


I suspect there are examples of non totally geodesic minimal submanifolds of any codimension between $1$ and $n-2$ in any compact manifold $M$ of dimension bigger than $3$ but I don't know a way to prove it on top of my head. However the construction I can think of may produce singularities, I am not sure how one can avoid that.


