  We derive a dimensionally-reduced limit theory for an $n$-dimensional
nonlinear elastic body that is slender along $k$ dimensions. The starting point
is to view an elastic body as an $n$-dimensional Riemannian manifold together
with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional
Euclidean space. The equilibrium configuration is the immersion that minimizes
the average discrepancy between the induced and intrinsic metrics. The
dimensionally reduced limit theory views the elastic body as a $k$-dimensional
Riemannian manifold along with an isometric $W^{2,2}$-immersion in
$n$-dimensional Euclidean space and linear data in the normal directions. The
equilibrium configuration minimizes a functional depending on the average
covariant derivatives of the linear data. The dimensionally-reduced limit is
obtained using a $\Gamma$-convergence approach. The limit includes as
particular cases plate, shell, and rod theories. It applies equally to
"standard" elasticity and to "incompatible" elasticity, thus including as
particular cases so-called non-Euclidean plate, shell, and rod theories.
