We prove a general result about the short-time existence and uniqueness of second-order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in the literature, as the transverse Ricci flow, the Sasaki–Ricci flow and the SasakiJ-flow and motivate the study of other evolution equations. We also introduce a transverse version of the Kähler–Ricci flow adapting some classical results to the foliated case.
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