Abstract
Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.
Highlights
Generalized geometry, building on the work of Hitchin [9] and Gualtieri [7] and the structure of Courant algebroids, constitutes a rich mathematical environment
We define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow
We review the framework of the generalized Ricci flow first introduced in [5, 22] and later described and studied in [6] by Garcia-Fernandez and Streets
Summary
Generalized geometry, building on the work of Hitchin [9] and Gualtieri [7] and the structure of Courant algebroids, constitutes a rich mathematical environment. This condition generalizes the Ricci soliton condition Rcg = λg + LX g, where Rcg denotes the Ricci tensor of g, λ ∈ R and LX g denotes the Lie derivative of g with respect to a vector field X. We define a family of flows of such structures, showing that they generalize the constructions known in literature as bracket flows, which have been extensively used to rephrase geometric flows on (nilpotent) Lie groups (see, for example, [15]).
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