Ricci-flat manifold

In mathematics, Ricci-flat manifolds^[1]^[2] are Riemannian manifolds whose Ricci curvature vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish.

Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space. For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature.

Ricci-flat manifolds often have restricted holonomy groups. Important cases include Calabi–Yau manifolds and hyperkähler manifolds.

Applications

In physics, Ricci-flat manifolds represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant.

References

↑ Dictionary of Distances By Michel-Marie Deza, Elena Deza. Elsevier, Nov 16, 2006. Pg 87
↑ Arthur E. Fischer and Joseph A. Wolf, The structure of compact Ricci-flat Riemannian manifolds. J. Differential Geom. Volume 10, Number 2 (1975), 277-288.

String theory
Background	Strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism
String duality	T-duality S-duality U-duality Montonen–Olive duality
Particles and fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂, F₄, E₆, E₇, E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Nima Arkani-Hamed Michael Francis Atiyah Tom Banks David Berenstein Raphael Bousso Gerald Cleaver Thomas Curtright Robbert Dijkgraaf Jacques Distler Michael Douglas Michael Duff Sergio Ferrara Willy Fischler Daniel Friedan Sylvester James Gates Ferdinando Gliozzi Rajesh Gopakumar Michael Green Brian Greene David Gross Steven Gubser Sergei Gukov Jeffrey Harvey Petr Hořava Gary Gibbons Shamit Kachru Michio Kaku Renata Kallosh Theodor Kaluza Anton Kapustin Igor Klebanov Vadim Knizhnik Maxim Kontsevich Oskar Klein Andrei Linde Juan Maldacena Stanley Mandelstam Donald Marolf Emil Martinec Shiraz Minwalla Gregory Moore Luboš Motl Sunil Mukhi Robert Myers Dimitri Nanopoulos Asad Naqvi Horațiu Năstase Nikita Nekrasov André Neveu Holger Bech Nielsen Peter van Nieuwenhuizen Sergei Novikov David Olive Hirosi Ooguri Joseph Polchinski Alexander Polyakov Arvind Rajaraman Pierre Ramond Lisa Randall Seifallah Randjbar-Daemi Martin Roček Ryan Rohm Joël Scherk John H. Schwarz Nathan Seiberg Ashoke Sen Steve Shenker Warren Siegel Eva Silverstein Đàm Thanh Sơn Matthias Staudacher Andrew Strominger Raman Sundrum Leonard Susskind Gerard 't Hooft Paul Townsend Sandip Trivedi Cumrun Vafa Gabriele Veneziano Erik Verlinde Herman Verlinde Julius Wess Edward Witten Shing-Tung Yau Tamiaki Yoneya Alexander Zamolodchikov Alexei Zamolodchikov Eric Zaslow Bruno Zumino Barton Zwiebach

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Ricci-flat manifold

Applications

Further reading

See also

References