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The study of calculus on differentiable manifolds is known as differential geometry. "Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth (which itself has many meanings), and analytic.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra .
Sjamaar, Reyer (2006), Manifolds and differential forms lecture notes (PDF), a course taught at Cornell University. Bachman, David (2003), A Geometric Approach to Differential Forms, arXiv: math/0306194, Bibcode:2003math.....6194B, an undergraduate text. Needham, Tristan. Visual differential geometry and forms: a mathematical drama in five acts ...
If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion f : M → N in general position the set of k-tuple points is an (n − k(n − m))-dimensional manifold. Every embedding is an immersion without multiple points (where k > 1). Note, however, that the converse is false: there are injective immersions that ...
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space , the n {\displaystyle n} -sphere , hyperbolic space , and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids , are all examples of ...
Spivak, Michael (1999) A Comprehensive Introduction to Differential Geometry (3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels.
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.