Noncommutative Einstein manifolds

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Our modest aim is to give a reasonable generalization of this notion for the noncommutative setting along the lines sketched above: in noncommutative geometry the information about a space is basically encoded in a so–called spectral triple (A, H, D) consisting of an algebra A and an operator D, classically corresponding to the Dirac operator, both acting on a Hilbert space H2 and some further data and relations; a variation of the geometry corresponds to a certain variation of the spectral triple and a noncommutative Einstein space will be characterized as a2(D2), coming from the heat kernel asymptotics, being critical under particular variations.