We consider Kobayashi geodesics in the moduli space of abelian
varieties $A_g$, that is, algebraic curves that are totally geodesic
submanifolds for the Kobayashi metric. We show that Kobayashi
geodesics can be characterized as those curves whose logarithmic
tangent bundle splits as a subbundle of the logarithmic tangent
bundle of $A_g$.
¶ Both Shimura curves and Teichmöller curves are examples of
Kobayashi geodesics, but there are other examples. We show
moreover that non-compact Kobayashi geodesics always map to
the locus of real multiplication and that the $Q$-irreducibility of
the induced variation of Hodge structures implies that they are
defined over a number field.