Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
The above seems to be always locally true if $a,b,c$ are sufficiently close to each other, but not globally. Is it really?If it isn't, could you give a counterexample?
1) But is there a sufficient condition one can put on $S$ so that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)?$
2) How about a necessary and sufficient condition?