Differentiable monotone maps on manifolds

Notes on Differentiable Manifolds

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...

Differentiable piecewise-Bézier interpolation on Riemannian manifolds

We propose a generalization of classical Euclidean piecewiseBézier surfaces to manifolds, and we use this generalization to compute a C1-surface interpolating a given set of manifold-valued data points associated to a regular 2D grid. We then propose an efficient algorithm to compute the control points defining the surface based on the Euclidean concept of natural C2-splines and show examples o...

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...

Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve Cand C-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C-continuity cannot be ensured by simple con...