nLab Riemannian geometry

Idea

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.

Related concepts

• fundamental theorem of Riemannian geometry

• Nash embedding theorem

• Laplace operator, wave equation

• spectral geometry, hearing the shape of a drum, spectral triple

• pseudo-Riemannian geometry, gravity, spacetimes

• harmonic map

• calibrated manifold

curvature in Riemannian geometry
Riemann curvature
Ricci curvature
scalar curvature
sectional curvature
p-curvature

$\,$

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

In index theory:

• Stolz conjecture

References

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:

• Luther P. Eisenhart, Riemannian Geometry, Princeton University Press (1925, 1950) [ISBN:9780691023533, ark:/13960/t47q47r0k, pdf]

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:

• Zuoqin Wang, Riemannian geometry (2024) [webpage]

With focus on special holonomy:

• Simon Salamon, Riemannian Geometry and Holonomy Groups, Research Notes in Mathematics 201, Longman (1989)

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]