The course presents the abstract generalisation of many objects of geometry, linear algebra and differential calculus. Riemanian differential geometry is presented as a smart and powerful synthesis of elementary algebra, geometry and analysis.
The course is divided into four parts: (i) differentiable manifolds and tensor calculus (tensors and tensor fields , curves, surfaces and sub-manifolds, diffeomorphisms, flows and Lie derivatives) ; (ii) curvature and torsion on differentiable manifolds (parallel transport, affine connection, covariant derivative, curvature and torsion) ; (iii) riemanian and pseudo-riemanian geometry (metric and pseudo-metric, geodesics, Levi-Civita connection, Riemann tensor and applications) and (iv) diffenrential forms and exterior calculus (exterior derivative and exterior algebra, volume form, geometrisation of vector calculus, integration of differential forms and Cartan structure equations).
