  We study some particular loci inside the moduli space $\mathcal{M}_g$, namely
the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an
elliptic curve $E$) and the bihyperelliptic locus (i.e. the locus of curves
admitting a $2:1$ cover over a hyperelliptic curve $C'$, $g(C') \geq 2$). We
show that the bielliptic locus is not a totally geodesic subvariety of
$\mathcal{A}_g$ if $g \geq 4$ (while it is for $g=3$, see [16]) and that the
bihyperelliptic locus is not totally geodesic in $\mathcal{A}_g$ if $g \geq
3g'$. We also give a lower bound for the rank of the second gaussian map on the
generic point of the bielliptic locus and an upper bound for this rank for
every bielliptic curve.
