Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

Andrijana Dekic; Marijana Babic; Srdjan Vukmirovic

Paper · Jun 14, 2021

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

Andrijana Dekic, Marijana Babic, Srdjan Vukmirovic

math.DGarXiv:2106.07320

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Abstract

It is well known that $C H^{n}$ has the structure of solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that all of these metrics are of constant negative scalar curvature and only one of them is Einstein (up to isometry and scaling). Finally, we present the relation between Ricci solitons on Heisenberg group and Einstein metric on $C H^{n}$ .

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Submitted: Jun 14, 2021
Subjects: math.DG

Ricci solitons Einstein metrics Scalar curvature Homogeneous spaces

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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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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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11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[1] 11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

[2] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[3] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

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