| A local invariant of Riemannian geometry is an assignment to each Riemannian manifold a function on the manifold. The value of an assigned function at a point is required to depend only on the geometry of the manifold in a neighborhood of the point. The local invariants in the thesis are given as polynomials in the derivatives of the metric. There is a natural order function on polynomial invariants. The main result in invariance theory in the thesis gives all p-form-valued polynomials in dimension m which have order m and which are zero on any product of the form MxS('1).;For certain elliptic partial differential operators, D, there are polynomial invariants a(,n)(D) called the invariants of the heat equation. These are defined in terms of the spectrum of D. It is, in general, very hard to compute any a(,n)(D). The thesis contains some new formulas giving certain linear combinations of the a(,n)(D) for operators D arising from the de Rham, the signature, and the spin complexes. |
