Notes on ‘Infinitesimal derivative of the Bott class and the Schwarzian derivatives’. (English) Zbl 1414.32029

Asuke, Taro

Tohoku Math. J. (2) 69, No. 1, 129-139 (2017).

The paper is a continuation of the author’s works in 2009 and in 2015. Namely, in [Tohoku Math. J. (2) 61, No. 3, 393–416 (2009; Zbl 1198.32014)], the author studied the infinitesimal derivative of the Bott class (in the complex case) which was defined by generalizing Heitsche’s construction and showed a beautiful formula which represents the infinitesimal derivative via the Schwarzian derivative. This formula was a generalization of Maszczyk’s one for the Godbillon-Vey class (in the real case) of real codimension-one foliations. In [Int. J. Math. 26, No. 4, Article ID 1540001, 29 p. (2015; Zbl 1325.58009)], the author proved that the infinitesimal derivatives of the Bott class (in the complex case) as well as those of the Godbillon-Vey class (in the real case) are determined by an invariant which was constructed by using the cohomology of foliations equipped with a transverse projective structure.
In this paper, the author introduces a new proof of the formula representing the infinitesimal derivatives of the Bott class and Godbillon-Vey class via the projective Schwarzian derivatives by using the calculations in [loc. cit.] for the coefficients of the transverse Thomas-Whitehead projective connections on the considered foliation.

Reviewer: Le Anh Vu (Ho Chi Minh City)

MSC:

32S65	Singularities of holomorphic vector fields and foliations
58H10	Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
53B10	Projective connections
58H15	Deformations of general structures on manifolds
53C12	Foliations (differential geometric aspects)
57R30	Foliations in differential topology; geometric theory

Keywords:

infinitesimal derivative; Bott class; Godbillon-Vey class; Schwarzian derivative; Thomas-Whitehead projective connections

Citations:

Zbl 1198.32014; Zbl 1325.58009