Let M be a closed manifold, ρ a representation of π1(M) on an A-Hilbert model W of finite type (A a finite von Neumann algebra) and μ a Hermitian structure on the flat bundle E → M associated to ρ. The relative torsion, first introduced by Carey, Mathai and Mishchenko, [CMM], associates to any pair (g, τ), consisting of a Riemannian metric g on M and a generalized triangulation τ = (h, g), a nu...

S f t,,-,] \-fT,I f 2 t,,--fT,] The Schwarzian is important in many areas of complex analysis (see, for example, the recent book of O. Lehto ILl) but it occurs first and foremost through its connection with M6bius transformations. The basic facts are az+b (1.1) S(f) 0 if and only iff(z) ad bc :/: 0 cz+d’ and (1.2) S(fo h) S(h) if and only iff is M6bius. Equation (1.2) is a special case of a gen...

Our goal is to study quantities in Riemannian geometry which remain invariant under the “conformal change of metrics”–that is, under changes of metrics which stretch the length of vectors but preserve the angles between any pair of vectors. We call such a quantity “conformally invariant”. In conjunction with the study of conformal invariants, we are also interested in studying “conformally cova...

Li, Jiayu. M.S., Purdue University, May 2015. Tracking Sales Activities in Agribusiness. Major Professor: W. Scott Downey. Decisions in the sales area, including customer and product selection and margin discipline, shape profits for companies in agribusiness. Management of the sales function takes place at the organizational, managerial, and practitioner level, each of which requires data abou...

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show thatM is a hyperbolic manifold of constant sectional curvature −h 2 4 , provided M is asymptotically harmonic of constant h > 0.

We compute the functional determinant quotient (det P h)=(det P g) for the Paneitz operator P in conformally related Riemannian metrics g; h, and discuss related positivity questions.

for a convex, coercive continuous Hamiltonian on closed Riemannian manifold M, we construct unique forward weak KAM solution of $$H(x,{d_x}u) = c(H)$$ by vanishing discount approach, where c(H) is the Mañé critical value. We also discuss dynamical significance such special solution.

A connected Riemannian manifold (M,g) is said to be isotropy irreducible if for each point p ∈ M the isotropy group Hp, i.e. all isometries of g fixing p, acts irreducibly on TpM via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By Schur’s lemma the metric g is unique up to...

which is a symplectic analog of the well-known de Rham–Hodge ∗operator on oriented Riemannian manifolds: one should use the symplectic form instead of the Riemannian metric. Going further, one can define operator δ = ± ∗ d∗, δ = 0. The form α is called symplectically harmonic if dα = 0 = δα. However, unlike de Rham–Hodge case, there exist simplectically harmonic forms which are exact. Because o...