Zoran Skoda differential geometry

For a radically modern point of view see parts of the nlab entry differential geometry.

Differential geometry is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces…) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional differentiable manifolds modelled locally on topological vector spaces. Techniques of differential calculus can be further stretched to generalized smooth spaces. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, togehter with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called differential topology.

Modern subfields of differential geometry include symplectic geometry, contact geometry, Riemannian geometry, Finsler, pseudoriemannian, symmetric spaces, Fréchet manifolds. There are alterantive foundations possible, for example the synthetic differential geometry.

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