Differential Geometry of Microlinear Frolicher Spaces I

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher 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 Frolicher spaces. Therein we have shown that Frolicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Frolicher spaces are the central object of infinite-dimensional differential geometry, we develop the theory of vector fields on them in this paper. The central result is that all vector fields on such a Frolicher space form a Lie algebra.