Syllabus
Tensors vector space and its dual, tensor product, multi-linear tensor maps, transformation of components, tensor notation
Manifolds basic notion of topology, differential structure, tangent spaces, vector and tensor fields, Lie brackets
Mappings of manifolds and Lie derivative mappings of manifolds, induced map, diffeomorphism, flow, Lie derivative
Exterior calculus wedge product, exterior derivative, exact and closed forms
Riemann and pseudo-Riemann geometry metric, signature, length of curves and distance, Hodge dual, Levi-Civita tensor, coderivative
Covariant derivative parallel transport, covariant derivative, covariant differential, geodesics, normal coordinates; torsion, Riemann curvature tensor, commutator of covariant derivatives for scalars and general tensors, Bianchi identities, Ricci tensor
Space of covariant derivatives pseudo-derivative, difference of two connections and differential tensor, coordinate derivative, n-ade derivative, Ricci (spin) coefficients, metric derivatives, contorsion tensor
Levi-Civita covariant derivative uniqueness, Christoffel symbols, Cartan structure equations, irreducibile splitting of Riemann tensor, Weyl tensor, scalar curvature, Einstein tensor
Relations between Lie, exterior and covariant derivatives
Lie and exterior derivative in terms of covariant derivative; Killing vectors and symmetries
Submanifolds and distributions immersion and embedding, adjusted coordinates, tangent and normal spaces; distributions, integrability conditions, Frobenius theorem
Integration on manifolds integrable densities, relation to anti-symmetric forms, integration of forms and densities; tensor of orientation, density dual, metric density; divergence of tensor densities, covariant derivative of densities, derivative annihilating density
Integral theorems generalized Stokes' theorem for forms, normal and tangent restriction of tensor densities, Stokes and Gauss theorems
Annotation
Foundations of topology; differentiable manifolds, tangent bundles, vector and tensor fields; maps of manifolds, diffeomorphism, induced mapping, Lie derivative; exterior calculus; covariant derivative, parallel transfer and geodesic curves, torsion and curvature, space of connections; (pseudo-)Riemann manifolds, metric derivatives,
Levi-Civita derivative, Killing vectors; integrability and Frobenius theorem; integration on manifolds, integrable densities, integral theorems.