Paper · Jun 29, 2016

Ricci solitons on low-dimensional generalized symmetric spaces

Giovanni Calvaruso, Eugenia Rosado

math.DGarXiv:1606.09166
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Abstract

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system, we prove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.

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Submitted
Jun 29, 2016
Subjects
math.DG
DOI
10.1016/j.geomphys.2016.11.008
Ricci solitons3-manifolds4-manifoldsHomogeneous spaces

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  1. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “pseudo Riemannian”, “algebraic Ricci”, “four dimensional”

  2. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim

    Topics 4-manifolds, 3-manifolds, Homogeneous spaces · Same author Giovanni Calvaruso · Shared terms “four dimensional”

  3. Solvable Algebraic Generalized Ricci Solitons in low dimensions

    Lorenzo Mucciante

    Topics 4-manifolds, 3-manifolds, Homogeneous spaces · Shared terms “algebraic”, “four dimensional”

  4. Evolution and metric signature change of maximally symmetric spaces under the Ricci flow

    R. Cartas-Fuentevilla, A. Herrera-Aguilar and J. A. Olvera-Santamaría

    Topics 4-manifolds, 3-manifolds · Shared terms “symmetric spaces”, “pseudo Riemannian”, “signature”

  5. Four-dimensional Lorentzian algebraic Ricci solitons

    Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “algebraic Ricci”, “four dimensional”

  6. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics 4-manifolds, Homogeneous spaces · Shared terms “dimensional generalized”, “four dimensional”

  7. Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

    Steven Gindi

    Topics 4-manifolds, 3-manifolds, Homogeneous spaces · Shared terms “four dimensional”

  8. Ricci Solitons on Pseudo-Riemannian Hypersurfaces of 4-dimensional Minkowski space

    Burcu Bektaş Demirci

    Topics 4-manifolds, Ricci solitons · Shared terms “pseudo Riemannian”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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