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:
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]
Monographs:
John M. Lee, Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics 176 (1997), Springer. ISBN: 0-387-98271-X.
Second Edition (retitled): Introduction to Riemannian Manifolds (2018), Springer. 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)
Marcel Berger, A panoramic view of Riemannian geometry
With an eye towards application 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 March 12, 2024 at 14:19:52. See the history of this page for a list of all contributions to it.