Vertical Morse-Bott-Smale Flows and Characteristic Forms

We present here several applications to the general homotopy formula found in [5]. We answer a question of Quillen about the degeneracy of a 1-parameter family of Chern character forms induced by superconnections associated to a Hermitian endomorphism. We prove a Poincaré Duality result about the transgression of the Pfaffian. We construct the Maslov spark, a de Rham-Federer differential character strongly related to the spectral flow of a family of unitary operators. Finally, we give a short proof of the classical Chern-Gauss-Bonnet theorem and of a result by Getzler.