Introduction to Differentiable Manifolds.

Notes on Differentiable Manifolds

Manifolds of Differentiable Densities

We develop a family of infinite-dimensional (i.e. non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class Ck b with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and the dually flat geometry of Amari...

Stable Equivalence of Differentiable Manifolds

A natural question, of great generality, various special forms of which are often asked in differential topology, is the following: Let Mi, Mi be differentiable w-manifolds, : M\—»M2 a continuous map which is a homotopy equivalence between M\ and M2. When is there a differentiable isomorphism <3>: Mi—> M2 in the same homotopy class as 0? For example, there is the Poincaré Conjecture which po...

Lecture Notes on Differentiable Manifolds

1. Tangent Spaces, Vector Fields in R and the Inverse Mapping Theorem 1 1.1. Tangent Space to a Level Surface 1 1.2. Tangent Space and Vectors Fields on R 2 1.3. Operator Representations of Vector Fields 3 1.4. Integral Curves 4 1.5. Implicitand Inverse-Mapping Theorems 5 2. Topological and Differentiable Manifolds 9 3. Diffeomorphisms, Immersions, Submersions and Submanifolds 9 4. Fibre Bundle...

AN INTRODUCTION TO THE THEORY OF DIFFERENTIABLE STRUCTURES ON INFINITE INTEGRAL DOMAINS

A special class of differentiable functions on an infinite integral domain which is not a field is introduced. Some facts about these functions are established and the special case of z is studied in more detail