Einstein Manifolds and Topology

A Note on Scalar Curvature Functions of Almost-Kähler Metrics

“…where r is the Ricci curvature tensor of g, ∆ g is the Laplacian, tr g (h) is the trace of h with respect to g, δ g h is the divergence of h which can be written in local coordinates as (δ g h) λ = −∇ ν h νλ and finally δ g (·) for 1-forms is the formal adjoint of the exterior differential on functions, [3].…”

“…Kazdan and Warner classified closed smooth manifolds of dimension> 2 into three classes according to what the scalar curvature functions can be on a manifold [3,Th. 4…”

SCALAR CURVATURES ON SU(3)/T(k, l)

“…Then the set of G-invariant symmetric covariant 2-tensor fields on G/H can be identified with the set of Ad(H)-invariant symmetric bilinear forms on m. In particular, the set of G-invariant Riemannian metrics on G/H is identified with the set of Ad(H)-invariant inner products on m (cf. [2,3,6,7]). …”

Gravitational dressing of D-instantons

“…Moreover, in four real dimensions, the hyper-Kähler property of the metric is equivalent to Anti-Self-Duality (ASD) of its Riemann curvature tensor [16]. As regards four-dimensional Kähler metrics, their Riemann ASD members are just Ricci-flat [17].…”

“…, where (T a ) k n is the NLSM torsion [16]. This torsion is induced by 4d gravity because the quaternionic condition on the hypermultiplet NLSM metric is the direct consequence of local N=2 supersymmetry in four spacetime dimensions [6].…”