Overdetermined PDE Systems on Some Classes of Riemannian Manifolds

Maryam Samavaki, Jukka Tuomela

Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101, Joensuu, Finland.

Abstract: We study several classes of Riemannian manifolds which are dened by imposing a certain condition on the Ricci tensor. We consider the following cases: Ricci recurrent, Cotton, quasi Einstein, and pseudo Ricci symmetric condition. Such conditions can be interpreted as overdetermined PDE systems whose unknowns are the components of the Riemannian metric, and perhaps, in addition, some auxiliary functions. Hence even if the dimension of the manifold is small it is not easy to compute interesting examples by hand, and indeed very few examples appear in the literature. We will present large families of nontrivial examples of such manifolds. The relevant PDE systems are rst transformed into an involutive form. After that in many cases, one can actually solve the resulting system explicitly. However, the involutive form itself already gives a lot of information about the possible solutions to the given problem. We will also discuss some relationships between the relevant classes.

Keywords: Cotton tensor, conformally conservative manifold, pseudo Ricci symmetric manifold, quasi Einstein manifold, Ricci recurrent manifold, overdetermined PDE.

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