Let $(M,g)$ be a Riemannian manifold of dimension $n$. (In the case I am interested in, $M$ is a complex symmetric domain, but I do not think that this is relevant for the question.)


Let $N$ be a submanifold of $M$. I would like to say that $N$ is "much curved". For the time being, let me just say that the second fundamental form - aka shape tensor - of $N$ has maximal rank. Does this imply that $N$ does not contain any geodesic segment? More generally, do I get some sort of lower bound on the curvature of curves contained in $N$?


The answer is likely to be negative I think, even if I do not have a countrexample. Let me ask a more vague question. Is there a notion of curvature for $N$ which prevent it from containing geodesic segments? Or, more generally, to bound the shape tensor of curves contained in $N$?



Curvature and intersection of submanifolds
If you assume that the codimension of $N$ is large, 
then your embeddings are called "free" (see the Gromov's book on h-principle). In this case there are no geodesics and all the curves in $N$ have non-vanishing normal curvature. 


For small codimension, the statement does not hold, say consider the paraboloid $z=xy$. 
(I do not see what happens in the intermediate codimension.)


