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.
"Kobayashi geodesics in $A_g$." J. Differential Geom. 86 (2) 355 - 379, October 2010. https://doi.org/10.4310/jdg/1299766791