differential geometry
Noun
Meaning
differential geometry (usually uncountable, plural differential geometries)
(geometry, mathematical analysis) The study of geometry, especially geometric structures on differentiable manifolds, using techniques from calculus, linear algebra and multilinear algebra.
Synonyms
Examples
The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves.
An advanced textbook on Clifford algebras and their applications to differential geometry.
Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds that have only trivial local moduli ; differential geometry, at least one non-trivial local moduli.
A. Gullstrand (vide supra, and Ann. d. Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.
From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut.
Geometric calculus Geometric calculus extends the formalism to include differentiation and integration including differential geometry and differential forms.