EQUIVARIANT TENSORS ON POLAR MANIFOLDS Ricardo Mendes Wolfgang Ziller, Advisor This PhD dissertation has two parts, both dealing with extension questions for equivariant tensors on a polar G-manifold M with section Σ ⊂ M. Chapter 3 contains the first part, regarding the so-called smoothness conditions: If a tensor defined only along Σ is equivariant under the generalized Weyl group W (Σ), then it exends to a G-equivariant tensor on M if and only if it satisfies the smoothness conditions. The main result is stated and proved in the first section, and an algorithm is also provided that calculates smoothness conditions. Chapter 4 contains the second part, which consists of a proof that every equivariant symmetric 2-tensor defined on the section of a polar manifold extends to a symmetric 2- tensor defined on the whole manifold. This is stated in detail in the first section, with proof. The main technical result used, called the Hessian Theorem, concerns the Invariant Theory of reflection groups, and is possibly of independent interest.