On Einstein Matsumoto metrics
Yi-Bing Shen, Xiaoling Zhang

TL;DR
This paper characterizes when Matsumoto metrics are Einstein, showing that under certain conditions, they are Ricci-flat and have parallel 1-forms, with explicit examples provided.
Contribution
It provides necessary and sufficient conditions for Matsumoto metrics to be Einstein, including a characterization involving Ricci-flatness and parallelism, and constructs explicit examples.
Findings
Matsumoto metrics are Einstein iff α is Ricci-flat and β is parallel when β's length is constant.
A nontrivial Ricci-flat Matsumoto metric example is constructed.
Conditions for Einstein Matsumoto metrics are fully characterized.
Abstract
In this paper, the necessary and sufficient conditions for Matsumoto metrics to be Einstein are given. It is shown that if the length of with respect to is constant, then the Matsumoto metric is an Einstein metric if and only if is Ricci-flat and is parallel with respect to . A nontrivial example of Ricci flat Matsumoto metrics is given.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Geometry and complex manifolds
