This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.

Contact metric manifolds satisfying a nullity condition

We prove that in E3 the Busemann-Petty problem, concerning cen- tral sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.

/pdf/a-positive-answer-to-the-busemann-petty-problem-in-three-1prwazgliu.pdf

A positive answer to the Busemann-Petty problem in three dimensions

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$
 -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$
 -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.

Canonical connections on paracontact manifolds

The purpose of this article is to survey the recent progress made on estimating positive eigenvalues of Laplacian on hyperbolic Riemann surfaces in the case of congruence subgroups in connection with the Selberg conjecture, as well as certain related ones. The results are obtained as consequences of establishing certain important cases of Langlands’ functoriality conjecture.

Surveys in differential geometry

It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in ¿-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d < 4 , and that suitably smooth axis-convex bodies of revolution are intersection bodies when d < 4. These results show that the Busemann-Petty problem has a negative answer for d > 5 and a positive answer for d = 3 and d = 4 when the body with smaller sections is a body of revolution.

https://cedar.wwu.edu/cgi/viewcontent.cgi?article=1027&context=math_facpubs

Intersection bodies and the Busemann-Petty problem

Soit (M, η,g) une variete de Riemann de contact de courbure φ-sectionnelle constante H. Alors les courbures de Ricci satisfont Ric(X,X)+Ric(φX, φX)≤3n−1+(n+1)H pour chaque vecteur unite X∈T x M x∈M, tels que η(X)=0. L'egalite est vraie pour tout x∈M et pour un vecteur unite X∈T x M tel que η(X)=0, si et seulement si (M, η, g) est sasakienne

/pdf/ricci-curvatures-of-contact-riemannian-manifolds-2m4c4fqy4d.pdf

Ricci curvatures of contact riemannian manifolds

/pdf/the-standard-cr-structure-on-the-unit-tangent-bundle-1wup46yzve.pdf

The standard {CR} structure on the unit tangent bundle

A characterization of the canonical spheres by the spectrum

/pdf/l-2-harmonic-forms-and-stability-of-minimal-hypersurfaces-2bgzwdmky7.pdf

$L^{2}$ harmonic forms and stability of minimal hypersurfaces

/pdf/remarks-on-a-triple-of-k-contact-structures-3twr24j4cv.pdf

Shukichi Tanno

Papers

Ricci curvatures of contact riemannian manifolds

The standard {CR} structure on the unit tangent bundle

A characterization of the canonical spheres by the spectrum

$L^{2}$ harmonic forms and stability of minimal hypersurfaces

Remarks on a triple of $K$-contact structures