This note covers
the following topics: Matrix Exponential; Some Matrix Lie Groups, Manifolds and
Lie Groups, The Lorentz Groups, Vector Fields, Integral Curves, Flows,
Partitions of Unity, Orientability, Covering Maps, The Log-Euclidean Framework,
Spherical Harmonics, Statistics on Riemannian Manifolds, Distributions and the
Frobenius Theorem, The Laplace-Beltrami Operator and Harmonic Forms, Bundles,
Metrics on Bundles, Homogeneous Spaces, Cli ord Algebras, Cli ord Groups, Pin
and Spin and Tensor Algebras.
This note explains the following
topics: From Kock–Lawvere axiom to microlinear spaces, Vector
bundles,Connections, Affine space, Differential forms, Axiomatic structure of
the real line, Coordinates and formal manifolds, Riemannian structure,
Well-adapted topos models.
This book is
addressed to the reader who wishes to cover a greater distance in a short time
and arrive at the front line of contemporary research. This book can serve as a
basis for graduate topics courses. Exercises play a prominent role while
historical and cultural comments relate the subject to a broader mathematical
context.
This
note contains on the following subtopics of Differential Geometry,
Manifolds, Connections and curvature, Calculus on
manifolds and Special topics.