  We prove that any smooth volume-preserving action of a lattice $\Gamma$ in
$\textrm{SL}(n,\mathbb{R})$, $n\ge 3$, on a closed $n$-manifold which possesses
one element that admits a dominated splitting should be standard. In other
words, the manifold is the $n$-flat torus and the action is smoothly conjugate
to an affine action. Note that an Anosov diffeomorphism, or more generally, a
partial hyperbolic diffeomorphism admits a dominated splitting. We have a
topological global rigidity when $\alpha$ is $C^{1}$. Similar theorems hold for
an action of a lattice in $\textrm{Sp}(2n,\mathbb{R})$ with $n\ge 2$ and
$\textrm{SO}(n,n)$ with $n\ge 5$ on a closed $2n$-manifold.
