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

/pdf/geometry-of-manifolds-which-admit-conservation-laws-ueh0lb9sc7.pdf

Geometry of manifolds which admit conservation laws

Conformal maps in Euclidean space

Recently I. Castro and F. Urbano introduced the Lagrangian catenoid. Topologically, it is {mathbb R} \times Sn - 1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in {mathbb C}n is foliated by round (n - 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in {mathbb C}n. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

On Lagrangian Catenoids

We give a classification of real hypersurfaces in complex proiective space under assumptions that the structure vector t is principal, the focal map has constant rank and that V f = o, where C is the Weyl conformal curvature tensor of the real hvoersurface.

/pdf/on-a-type-of-real-hypersurfaces-in-complex-projective-space-1w0njsmqcn.pdf

On a type of real hypersurfaces in complex projective space

While the study of complex contact manifolds is almost as old as the modern theory of real contact manifolds, the subject has received much less attention, and since many examples are now appearing in the literature, we devote this and the next chapter to the subject.

David E. Blair

Papers

Geometry of manifolds which admit conservation laws

Conformal maps in Euclidean space

On Lagrangian Catenoids

On a type of real hypersurfaces in complex projective space

Complex Contact Manifolds