Let $V_1,\ldots,V_k$ be a transversal set of smooth compact orientable sub-manifolds of a compact orientable manifold $M$, and set $V=\bigcap V_i$.

Is it always possible to equip a neighborhood $U$ of $V$ in $M$ with a metric $g$ such that every $V_i\cap U$ is totally geodesic?

or that the exponential map $exp: N_V^M=TV^{\perp g} \to U$ maps very $(N_V^M \cap TV_i)|_V$ into $V_i$?

If not, what are the obstructions?

Edit (Misha): Just to clarify things: Smooth submanifolds $M_1,...,M_k$ of a smooth manifold $M$ are said to *intersect transversally* at a point $x$ if
$$
codim_{T_x M} \bigcap_{i=1}^k T_x M_i= \sum_{i=1}^k codim_M M_i.
$$
In other words, codimension of intersection (at the tangent space level) is the sum of codimensions.