synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In differential geometry, by a current on a smooth manifold one means a generalized function/distribution with values in differential forms.
See current and conserved current for other notions of “currents”.
The original article:
Textbook accounts:
Herbert Federer, Section 4.1 in: Geometric measure theory, Grundlehren 153, Springer (1969, 1996) [doi:10.1007/978-3-642-62010-2]
Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys and Monographs 14, AMS (1977) [ams:surv-14]
Georges de Rham, Chapter III of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]
Steven G. Krantz, Harold Parks, Section 7 of: Geometric integration theory, Springer (2008) [pdf]
Frank Morgan, Geometric measure theory: a beginner’s guide, Academic Press (2016) [doi:10.1016/C2015-0-01918-9]
See also:
Currents appear as part of “geometric cycles” for differential cobordism cohomology by actual cobordism-classes equipped with differential geometric data:
and in the Hodge-filtered version:
Knut Bjarte Haus, §2.3, §4.12 in: Geometric Hodge filtered complex cobordism, PhD thesis (2022) [ntnuopen:3017489]
Knut Bjarte Haus, Gereon Quick, §2 of: Geometric Hodge filtered complex cobordism [arXiv:2210.13259]
Last revised on June 10, 2023 at 07:55:03. See the history of this page for a list of all contributions to it.