IJPAM: Volume 64, No. 1 (2010)

DIFFERENTIAL GEOMETRY OF
MICROLINEAR FRÖLICHER SPACES I

Hirokazu Nishimura
Institute of Mathematics
University of Tsukuba
Tsukuba, Ibaraki, 305-8571, JAPAN


Abstract.The central object of synthetic differential geometry is microlinear spaces. In our previous paper (Microlinearity in Frölicher spaces - beyond the regnant philosophy of manifolds, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24) we have emancipated microlinearity from within well-adapted models to Frölicher spaces. Therein we have shown that Frölicher spaces which are microlinear as well as Weil exponentiable form a Cartesian closed category. To make sure that such Frölicher spaces are the central object of infinite-dimensional differential geometry, we develop the theory of vector fields on them in this paper. Our principal result is that all vector fields on such a Frölicher space form a Lie algebra.

Editorial Remark. Please, see International Journal of Pure and Applied Mathematics, 65, No. 1 (2010), here.

Received: June 30, 2010

AMS Subject Classification: 58A03

Key Words and Phrases: microlinearity, synthetic differential geometry, Frölicher space, transversal limit diagram, Topos theory, infinite-dimensional differential geomery, Cahiers Topos, Cartesian closed category, Weil algebra, Weil functor, nilpotent infinitesimal

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 64
Issue: 1