We consider the heat kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented.

Functional determinants on various domains of the sphere and flat space are presented for scalar and spinor fields.

The mode properties for spectral and mixed boundary conditions for massless spin- fields are derived for the d-ball. The corresponding functional determinants and traced heat-kernel coefficients are presented, the latter as polynomials in d.

Functional determinants for the scalar Laplacian on spherical caps and slices, flat balls, shells and generalized cylinders are evaluated in two, three and four dimensions using conformal techniques. Both Dirichlet and Robin boundary conditions are allowed for. Some effects of non-smooth boundaries are discussed; in particular, the 3-hemiball and the 3-hemishell are considered. The edge and vertex contributions to the coefficient are examined.

We obtain an expression for the curvature of the Lie group SDiff$\cal M$ and use it to derive Lukatskii's formula for the case where $\cal M$ is locally Euclidean. We discuss qualitatively some previous findings for SDiff$S^{2}$ in conjunction with our result.

Some calculational errors in expressions derived previously by the first author for the effective action, or equivalently for the functional determinant, on sectors of a spherical cap are corrected. The formula for the change in the effective action under Weyl rescalings in the three dimensional case is also amended.

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.