A new infinite series of Einstein metrics is constructed explicitly on S 2 × S 3 , and the non-trivial S 3 -bundle over S 2 , containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr black hole. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S d−2 -bundle over S 2 from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on CP 2 ♯CP
We consider Kerr-AdS black holes with equal angular momenta in arbitrary odd spacetime dimensions ≥ 5. Twisting the Killing vector fields of the black holes, we reproduce the compact Sasaki-Einstein manifolds constructed by Gauntlett, Martelli, Sparks and Waldram. We also discuss an implication of the twist in string theory and M-theory.
Abstract. We give a simple proof of Macdonald's formula for the volume of a compact Lie group.
Mathematics Subject Classification (1991). 22E15.
Keywords. Compact Lie group, Todd class.It is well-known that a compact Lie group G of rank r is rational homotopy equivalent to a product of spheresMacdonald calculated the volume of G with respect to the Haar measure µ induced from a Lebesgue measure λ on the Lie algebra g of G. The result is as follows:where g Z is the Chevalley lattice and σ is the product of the superficial measure of the unit spheres S 2mi+1 in R 2mi+2 , that isIn this note we shall give a simpler proof of this formula. Let T be a maximal subgroup of G, t its Lie algebra and t Z a lattice in t such that the kernel of the exponential map from t to T is 2πt Z . We fix a positive definite inner product on g which is invariant under the adjoint action of G such that the induced volume form coincides with λ. This inner product induces Lebesgue measures on t and g/t (also denoted by λ), which determine a Haar measure on T and a G-invariant measure on the flag manifold G/T (also denoted by µ). Then clearly µ(T ) = (2π) r λ(t/t Z ), µ(G) = µ(T )µ(G/T ).
The Ashtekar-Mason-Newman equations are used to construct the hyperkähler metrics on four dimensional manifolds. These equations are closely related to anti self-dual Yang-Mills equations of the infinite dimensional gauge Lie algebras of all volume preserving vector fields. Several examples of hyperkähler metrics are presented through the reductions of anti self-dual connections. For any gauge group anti self-dual connections on hyperkähler manifolds are constructed using the solutions of both Nahm and Laplace equations.
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