Differential forms and the noncommutative residue

Double Conformal Invariants and the Wodzicki Residue

“…Then, as in [Co] or [Ug1], we have Theorem 2.1. Ω g1,g2 d (f 1 , f 2 ) is a unique defined by (2.8), symmetric and double conformally invariant ndifferential form.…”

“…Let * be the Hodge star operator associated to the rescaling metric. By [Ug1], we have * = e (2p−n)f * . In particular, when 2p = n, the * operator is conformal invariant, that is * = * .…”

“…Connes also gave an explanation of the Polyakov action and its 4-dimensional analogy by using his conformal invariant. In [Ug1], Connes' result was generalized to the higher dimensional case and an explicit expression of the Connes' invariant in the flat case was given by Ugalde. In addition, the way of computation in the general case was indicated.…”

The General Kastler-Kalau-Walze Type Theorems About Witten Deformation

“…Connes [4][5][6] 's research shows that non-commutative residues play an integral role in noncommutative geometry. The research work of Connes [4] and Ugalde [7] shows that non-commutative residues are closely related to conformal geometry. That is, noncommutative residues bridge conformal geometry and noncommutative geometry.…”

The perturbation of the de Rham Hodge Operator and the Kastler-Kalau-Walze type theorem for manifolds with boundary

“…Ackermann proved this theorem by using the heat kernel expansion, in [7]. The result of Connes was extended to the higher dimensional case [8]. Fedosov et al gave the definition about the noncommutative residues on Boutet de Monvel algebra [9].…”