A concise yet self-contained advent of the important thoughts of recent topology and differential geometry on a mathematical point is given in particular with purposes in physics in brain. All uncomplicated recommendations are systematically supplied together with sketches of the proofs of such a lot statements. soft finite-dimensional manifolds, tensor and external calculus working on them, homotopy, (co)homology concept together with Morse idea of serious issues, in addition to the idea of fiber bundles and Riemannian geometry, are handled. Examples from physics include topological fees, the topology of periodic boundary stipulations for solids, gauge fields, geometric stages in quantum physics and gravitation.
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