Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain

/pdf/a-class-of-almost-contact-riemannian-manifolds-1yxyggdui1.pdf

A class of almost contact riemannian manifolds

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG
2 and Spin(7).

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253076

Real Killing spinors and holonomy

http://www.dcs.gla.ac.uk/~muffy/papers/calder-kolberg-magill-reiff.pdf

Feature interaction: a critical review and considered forecast

We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form Open image in new window1,9−n×Mn with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M7 admits a G2-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity.

G-Structures and Wrapped NS5-Branes

Symmetries and φ-symmetric spaces

Defining a function on the set of all Riemannian metrics associated to a contact form on a compact manifold by taking the integral of the Ricci curvature in the direction of the characteristic vector field, it is shown that on a compact regular contact manifold the only critical points of this function are the metrics for which the characteristic vector field generates a group of isometrics.

Critical associated metrics on contact manifolds. II

In [4] Calabi and Eckmann showed that the product of two odd-dimensional spheres S x S (p,q > 1) is a complex manifold. As S x S is not Kaehlerian, the fundamental 2-form Ω of the Hermitian structure is not closed. However, dΩ does have a special form on S X S in fact, S x S admits two nonvanishing vector fields which are both Killing and analytic, and whose covariant forms determine Ω. Our purpose here is to study complex manifolds whose complex structures are similar to the complex structure on S X S. In § 1 we review the geometry of the Calabi-Eckmann manifolds. In § 2 we give some elementary properties of vector fields on a Hermitian manifold, and introduce the notion of a holomorphic pair of automorphisms and of a bicontact manifold. § 3 continues the author's paper [2] on the differential geometry of principal toroidal bundles for the present case. In § 4 we discuss bicontact manifolds and, in particular, the integrable distributions of a bicontact structure on a Hermitian manifold. Finally in § 5 we give some results on the curvatures of a Hermitian manifold admitting a holomorphic pair of automorphisms.

/pdf/geometry-of-complex-manifolds-similar-to-the-calabi-eckmann-52574qvzh9.pdf

David E. Blair

Papers

Symmetries and φ-symmetric spaces

Critical associated metrics on contact manifolds. II

Geometry of complex manifolds similar to the Calabi-Eckmann manifolds

On the geometric meaning of the Bochner tensor

When is the tangent sphere bundle conformally flat