  In this thesis we give obstructions for Drinfel'd twist deformation
quantization on several classes of symplectic manifolds. Motivated from this
quantization procedure, we further construct a noncommutative Cartan calculus
on any braided commutative algebra, as well as an equivariant Levi-Civita
covariant derivative for any non-degenerate equivariant metric. This
generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan
calculus on twist star product algebras. We prove that the Drinfel'd functor
leads to equivalence classes in braided commutative geometry and commutes with
submanifold algebra projection.
