Differential forms  continued (orientability, integration on manifolds, the Stokes theorem and its consequences) Tensor fields on manifolds and their properties (definition, operations on tensors, including symmetrization, antisymmetrization, tensor product, the Lie derivative) Affine connections and related issues (the torsion tensor, the curvature tensor, parallel transport of vectors, geodesics, covariant derivatives, geometrical meaning of the curvature tensor) Manifolds with the metric ((pseudo) Riemannian manifolds, LeviCivita connection, curvature tensor, Ricci tensor, scalar curvature, isometries and the Killing equation, integrating functions on manifold with a metric, the LeviCivita (pseudo)tensor, volume element, Hodge duality). Basics of the Lie groups theory (the definition of the Lie group, left and rightinvariant vector fields and differential forms and their properties, the Lie algebra and its relationship with the Lie group)


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