#### Date of Award

Spring 5-16-2011

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Graduate Group

Mathematics

#### First Advisor

Wolfgang Ziller

#### Abstract

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.

#### Recommended Citation

Mendes, Ricardo Augusto E., "Equivariant Tensors on Polar Manifolds" (2011). *Publicly Accessible Penn Dissertations.* Paper 312.

http://repository.upenn.edu/edissertations/312

*Computer scripts in the GAP language*