  We propose a general procedure to construct noncommutative deformations of an
embedded submanifold $M$ of $\mathbb{R}^n$ determined by a set of smooth
equations $f^a(x)=0$. We use the framework of Drinfel'd twist deformation of
differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006),
1883]; the commutative pointwise product is replaced by a (generally
noncommutative) $\star$-product determined by a Drinfel'd twist. The twists we
employ are based on the Lie algebra $\Xi_t$ of vector fields that are tangent
to all the submanifolds that are level sets of the $f^a$; the twisted Cartan
calculus is automatically equivariant under twisted tangent infinitesimal
diffeomorphisms. We can consistently project a connection from the twisted
$\mathbb{R}^n$ to the twisted $M$ if the twist is based on a suitable Lie
subalgebra $\mathfrak{e}\subset\Xi_t$. If we endow $\mathbb{R}^n$ with a metric
then twisting and projecting to the normal and tangent vector fields commute,
and we can project the Levi-Civita connection consistently to the twisted $M$,
provided the twist is based on the Lie subalgebra
$\mathfrak{k}\subset\mathfrak{e}$ of the Killing vector fields of the metric; a
twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can
be characterized in terms of generators and polynomial relations. We present in
some detail twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and
twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [these are
twisted (anti-)de Sitter spaces $dS_2,AdS_2$].
