  We prove that certain Riemannian manifolds can be isometrically embedded
inside Calabi-Yau manifolds. For example we prove that given any real-analytic
one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold
$Y$ with volume form independent of $t$ and with a real-analytic family of
nowhere vanishing harmonic one forms $\theta_t$, then $(Y, g_t)$ can be
realized as a family of special Lagrangian submanifolds of a Calabi-Yau
manifold $X$. We also prove that certain principal torus bundles can be
equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus
action. We use this to construct examples of $n$-parameter families of special
Lagrangian tori inside $n+k$-dimensional Calabi-Yau manifolds with torus
symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian
fibrations with $T^2$-symmetry.
