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Register a new userEinstein Metrics, Projective Structures and the $SU(infty)$ Toda Equation
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We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [ abla])$. We show that there is a canonical Einstein of metric on an $R^*$--bundle over $T^*N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^*N$.
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We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective compactification of neutral signature Einstein metrics canonically defined on certain rank--$n$ affine bundles $M$ over $n$-dimensional manifolds endowed with projective structures.
We show how to lift solutions of Euclidean Einstein-Maxwell equations with non-zero cosmological constant to solutions of eleven-dimensional supergravity theory with non-zero fluxes. This yields a class of 11D metrics given in terms of solutions to $SU(infty)$ Toda equation. We give one example of a regular solution and analyse its supersymmetry. We also analyse the integrability conditions of the Killing spinor equations of N=2 minimal gauged supergravity in four Euclidean dimensions. We obtain necessary conditions for the existence of additional Killing spinors, corresponding to enhancement of supersymmetry. If the Weyl tensor is anti-self-dual then the supersymmetric metrics satisfying these conditions are given by separable solutions to the $SU(infty)$ Toda equation. Otherwise they are ambi-Kahler and are conformally equivalent to Kahler metrics of Calabi type or to product metrics on two Riemann surfaces.
Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koisos Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koisos. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R-plus is negative definite, the Hessian of S is strictly positive modulo gauge.
In this paper, the necessary and sufficient conditions for Matsumoto metrics $F=frac{alpha^2}{alpha-beta}$ to be Einstein are given. It is shown that if the length of $beta$ with respect to $alpha$ is constant, then the Matsumoto metric $F$ is an Einstein metric if and only if $alpha$ is Ricci-flat and $beta$ is parallel with respect to $alpha$. A nontrivial example of Ricci flat Matsumoto metrics is given.
We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Mreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.
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