Differential calculus for generalized geometry and geometric Lax flows

Shengda Hu

Paper · Jun 9, 2022

Differential calculus for generalized geometry and geometric Lax flows

Shengda Hu

math.DGarXiv:2206.04566v2

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Abstract

Employing a class of generalized connections, we describe certain differential complices $(Ω ~_{T}^{*} (M), d ~^{T})$ constructed from $\land^{*} T M$ and study some of their basic properties, where $T M = T M \oplus T^{*} M$ is the generalized tangent bundle on $M$ . A number of classical geometric notions are extended to $T M$ , such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when $T M$ is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenböck identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest.

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Submitted: Jun 9, 2022
Last revised: Jun 23, 2022 (v2)
Subjects: math.DG
Journal: Pacific J. Math. 331 (2024) 23-76
DOI: 10.2140/pjm.2024.331.23

Comments: 48 pages. Typos corrected and references updated

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Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[3] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

Differential calculus for generalized geometry and geometric Lax flows

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