Sasaki Metric Research Articles

Some Properties of Sasakian Metrics in Cotangent Bundles

The main aim of this paper is to investigate curvature properties of the Sasakian metrics in the cotangent bundle.

Stability of unit Hopf vector fields on quotients of spheres

The volume of a unit vector field V of a Riemannian manifold ( M , g ) is the volume of its image V ( M ) in the unit tangent bundle endowed with the Sasaki metric. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fiber of a Hopf fibration S n → C P n − 1 2 ( n odd) are well known to be critical for the volume functional on the round n-dimensional sphere S n ( r ) for every radius r > 1 . Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in Borrelli and Gil-Medrano (2006) [2], it is proven that for n ⩾ 5 there is a critical radius r c = 1 n − 4 such that Hopf vector fields are stable if and only if r ⩽ r c . In this paper we consider the question of the existence of a critical radius for space forms M n ( c ) ( n odd) of positive curvature c. These space forms are isometric quotients S n ( r ) / Γ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that r c = + ∞ , unless Γ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on S n ( r ) / Γ , Γ ≠ { Id } , whatever the radius.

On para-Nordenian structures

The aim of this paper is to investigate para-Nordenian properties of the Sasakian metrics in the cotangent bundle.

The Geometry of Generalised Cheeger-Gromoll Metrics

We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restrictions to the fibres have positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

NOTES ON TANGENT SPHERE BUNDLES OF CONSTANT RADII

We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M, g) of constant radius <TEX>$\gamma$</TEX> reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the <TEX>$\eta$</TEX>-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles.

A note of Sasakian metrics with constant scalar curvature

In this paper, we consider compact Sasakian manifolds with constant scalar curvature. Under some positive curvature assumption, we prove that such Sasakian metrics must be η-Einstein metrics.

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds.

Unit vector fields on real space forms which are harmonic maps

Gs), where e Gs is the Sasaki metric. In this paper, by using a different method, we get an analogue of Han and Yim’s theorem for a Riemannian three-manifold with constant sectional curvature k 6= 0. An immediate consequence is that there does not exist a unit vector field on the hyperbolic three-space that defines a harmonic map. We also extend this result for Riemannian (2n +1)-manifolds (M, g) of constant sectional curvature k > 0 with 1(M) 6= 0.

On the cohomology algebra of some classes of geometrically formal manifolds

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal K\"ahler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat. Finally we prove that a six-dimensional manifold with $b_1 \neq 1, b_2 \geqslant 2$ and not having the cohomology algebra of $\mathbb{T}^3 \times S^3$ carries a symplectic structure as soon as it admits a formal metric.

Sasaki–Einstein Manifolds and Volume Minimisation

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of $$\mathbb{R}$$ , with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S 1-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.

Highly connected manifolds with positive Ricci curvature

We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for computing the diffeomorphism types is given and tables are presented for dimensions 7 and 11.

General Kerr–NUT–AdS metrics in all dimensions

The Kerr–AdS metric in dimension D has cohomogeneity [D/2]; the metric components depend on the radial coordinate r and [D/2] latitude variables μi that are subject to the constraint ∑iμ2i = 1. We find a coordinate reparametrization in which the μi variables are replaced by [D/2] − 1 unconstrained coordinates yα, and having the remarkable property that the Kerr–AdS metric becomes diagonal in the coordinate differentials dyα. The coordinates r and yα now appear in a very symmetrical way in the metric, leading to an immediate generalization in which we can introduce [D/2] − 1 NUT parameters. We find that (D − 5)/2 are non-trivial in odd dimensions whilst (D − 2)/2 are non-trivial in even dimensions. This gives the most general Kerr–NUT–AdS metric in D dimensions. We find that in all dimensions D ⩾ 4, there exist discrete symmetries that involve inverting a rotation parameter through the AdS radius. These symmetries imply that Kerr–NUT–AdS metrics with over-rotating parameters are equivalent to under-rotating metrics. We also consider the BPS limit of the Kerr–NUT–AdS metrics, and thereby obtain, in odd dimensions and after Euclideanization, new families of Einstein–Sasaki metrics.

Kerr–de Sitter black holes with NUT charges

The four-dimensional Kerr–de Sitter and Kerr–AdS black hole metrics have cohomogeneity 2, and they admit a generalisation in which an additional parameter characterising a NUT charge is included. In this paper, we study the higher-dimensional Kerr–AdS metrics, specialised to cohomogeneity 2 by appropriate restrictions on their rotation parameters, and we show how they too admit a generalisation in which an additional NUT-type parameter is introduced. We discuss also the supersymmetric limits of the new metrics. If one performs a Wick rotation to Euclidean spacetime signature, these yield new Einstein–Sasaki metrics in odd dimensions, and Ricci-flat metrics in even dimensions. We also study the five-dimensional Kerr–AdS black holes in detail. Although in this particular case the NUT parameter is trivial, our investigation reveals the remarkable feature that a five-dimensional Kerr–AdS “over-rotating” metric is equivalent, after performing a coordinate transformation, to an under-rotating Kerr–AdS metric.

A topological minorization for the volume of vector fields on 5-manifolds

A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional. For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.

On Riemannian g-natural Metrics of the Form a.gs + b.gh + c.gv on the Tangent Bundle of a Riemannian Manifold (M, g)

Let (M, g) be a Riemannian manifold and TM its tangent bundle. In [5] we have investigated the family of all Riemannian g-natural metrics G on TM (which depends on 6 arbitrary functions of the norm of a vector u ∈ TM). In this paper, we continue this study under some additional geometric properties, and then we restrict ourselves to the subfamily {G=a.g s + b.g h + c.g v , a, b and c are constants satisfying a > 0 and a(a + c) − b2 > 0}. It is known that the Sasaki metric g s is extremely rigid in the following sense: if (TM, g s ) is a space of constant scalar curvature, then (M, g) is flat. Here we prove, among others, that every Riemannian g-natural metric from the subfamily above is as rigid as the Sasaki metric.

Powers of the curvature operator of space forms and geodesics of the tangent bundle

It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π 1463-01 Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k 3, k 6, and k 10 for the real, complex, and quaternionic space forms, respectively.

On the volume of unit vector fields on spaces of constant sectional curvature

A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres {\textbf {S}}^{2k+1} , this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.

CORRECTED ENERGY OF DISTRIBUTIONS ON RIEMANNIAN MANIFOLDS

In the mathematical literature there are several functionals which let us measure how the vector fields defined over any Riemannian manifold are ordered. We can ask ourselves which are the optimal vector fields. In fact, we try to measure how far from being parallel our vector field is. We can also extend this question to distributions. Gluck, Ziller [5] and Johnson [6], among others, studied the volume of unit vector fields. They define the volume of a unit vector field to be the volume of the submanifold in the unit tangent bundle defined by ( ). For this, we regard the vector field as a map : → 1 and in 1 we consider the Sasaki metric. We know [5] that in the ambient manifold S3 the Hopf vector fields, and no others, minimize this functional. For higher dimensional spheres, we know [6] that the Hopf vector fields are unstable critical points; that is, they are not even local minima. Wiegmink [8] defined the total bending of a unit vector field . This functional is related to the energy of the map : → 1 , as we shall see in Section 3. Brito [1] proved that the Hopf vector fields in S3 are the only minima of the total bending. Furthermore, he proved a more general result giving an absolute minimum in any dimension of the total bending corrected by the second fundamental form of the orthogonal distribution to the field . The coefficient of this correction vanishes in dimension 3 and then the corrected total bending agrees with the total bending. Similarly to the situation for vector fields, the energy of a -distribution V in a compact oriented Riemannian manifold is the energy of the section of the Grassmann manifold of -planes in induced by V . In this paper, we add to the energy the norms of the mean curvatures of V and its orthogonal distribution (with different weights) introducing in this way the corrected energy. In Theorem 1, we find a lower bound for the corrected energy of a foliation, and