3 - Fold Local index theorem

Zeta Forms and the Local Family Index Theorem

Let X be a C n-dimensional compact Riemannian manifold without boundary and let E = E ⊕ E be a Z2-graded (super) complex vector bundle over X. We write Γ(X,E) for the space of C sections of E and τ for the involution defining the induced Z2-grading. The super (or Z2-graded) trace of a trace class operator a on Γ(X,E) is defined by Str(a) = Tr (τa). Let A and F be classical (one-step polyhomogen...

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Zeta-chern Forms and the Local Families Index Theorem

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds π : M → B we show that there is a natural zeta-form ζθ(F, s) and zetadeterminant-form detζ,θ(F) in the de-Rham algebra of smooth differential forms A(B), generalizing the classical single operator spectral zeta function and determinant. I...

A Local - Global Theorem on Periodic Maps 3

Let ψ1, . . . , ψk be maps from Z to an additive abelian group with positive periods n1, . . . , nk respectively. We show that the function ψ = ψ1 + · · ·+ψk is constant if ψ(x) equals a constant for |S| consecutive integers x where S = {r/ns : r = 0, . . . , ns − 1; s = 1, . . . , k}; moreover, there are periodic maps f0, . . . , f|S|−1 : Z → Z only depending on S such that ψ(x) = |S|−1 r=0 fr...

A Local Slice Theorem for 3 - Dimensional Cr Structures