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)
Riemannian geometry studies smooth manifolds that are equipped with a Riemannian metric: Riemannian manifolds.
Riemannian geometry is hence equivalently the Cartan geometry for inclusions of the orthogonal group into the Euclidean group.
curvature in Riemannian geometry |
---|
Riemann curvature |
Ricci curvature |
scalar curvature |
sectional curvature |
p-curvature |
In index theory:
See also the references at differential geometry.
Riemannian geometry is named after:
Bernhard Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Göttingen (1845) [doi:10.1007/978-3-642-35121-1]
Engl. transl: William Clifford: On the hypotheses which underlie geometry, Nature VIII (1873) 183-184 [doi:10.1007/978-3-319-26042-6]
Early monograph:
Original discussion via Cartan geometry of coframe fields:
Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]
Élie Cartan (translated by Robert Hermann from Cartan’s lectures in 1951): Geometry of Riemannian Spaces, Lie Groups: History, Frontiers and Applications XIII, Math Sci Press (1983) [ark:/13960/s28rzmj9xrv]
Modern monographs:
Shoshichi Kobayashi, Katsumi Nomizu, Chapter IV onwards in: Foundations of differential geometry, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprint: Wiley Classics Library (1996) [ISBN:978-0-470-55558-3, Wikipedia entry]
William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press (1975, 1986), Elsevier (2002) [ISBN:9780121160517, pdf]
Thomas J. Willmore, Riemannian Geometry, Oxford University Press (1996) [ISBN:9780198514923, ark:/13960/t4jn0093w]
John M. Lee, Riemannian manifolds. An introduction to curvature, Graduate Texts in Mathematics 176 Springer (1997) [ISBN: 0-387-98271-X]
second edition (retitled):
John M. Lee, Introduction to Riemannian Manifolds, Springer (2018) [ISBN:978-3-319-91754-2, doi:10.1007/978-3-319-91755-9]
Isaac Chavel, Riemannian geometry – A modern introduction, Cambridge University Press (1993) [doi:10.1017/CBO9780511616822]
Marcel Berger, A panoramic view of Riemannian geometry, Springer (2003) [doi:10.1007/978-3-642-18245-7]
Peter Petersen: Riemannian Geometry, Springer (2006) [doi:10.1007/978-0-387-29403-2, pdf]
Lecture notes:
With focus on special holonomy:
With focus on pseudo-Riemannian manifolds and application to special and general relativity (gravity):
Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics 103, Academic Press (1983) [ISBN:9780125267403]
Shlomo Sternberg, Semi-Riemannian Geometry and General Relativity (2003) [pdf, ark:/13960/t5m927d2v]
Shlomo Sternberg, Curvature in Mathematical Physics, Dover (2012) [ISBN:9780486478555]
Generally with an eye towards applications in mathematical physics:
Mikio Nakahara, Chapter 6 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Gerd Rudolph, Matthias Schmidt, Chapter 2 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)
Jürgen Jost, Riemannian Geometry and Geometric Analysis, Springer (2017) [doi:10.1007/978-3-319-61860-9]
Last revised on July 30, 2024 at 13:02:55. See the history of this page for a list of all contributions to it.