Four-dimensional Riemannian Manifold Research Articles

Four-dimensional Riemannian product manifolds with circulant structures

"A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, is considered. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant, and it acts as an isometry with respect to the metric. The Riemannian product manifold associated with the considered manifold is studied. Conditions for the metric, which imply that the Riemannian product manifold belongs to each of the basic classes of Staikova-Gribachev's classi cation, are obtained. Examples of such manifolds are given."

On an Indefinite Metric on a Four-Dimensional Riemannian Manifold

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type (1,1), whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space.

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that mathbb {C}mathbb {P}^3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

Observational constraints on varying fundamental constants in a minimal CPC model

ABSTRACT A minimal model based on the Co-varying Physical Couplings (CPC) framework for gravity is proposed. The CPC framework is based on the assumptions of a metric-compatible four-dimensional Riemannian manifold, where a covariantly conserved stress-energy tensor acts as source of the field equations, which are formally the same as Einstein field equations, but where the couplings {G, c, Λ} are allowed to vary simultaneously. The minimal CPC model takes Λ as a genuine constant while c and G vary in an entangled way that is consistent with Bianchi identity and the aforementioned assumptions. The model is constrained using the most recent galaxy cluster gas mass fraction observational data. Our result indicates that the functions c(z) and G (z) = G0 (c/c0)4 are compatible with constant couplings for the two different parametrizations of c = c(z) adopted here.

Extrinsic curvature and conformal Gauss–Bonnet for four-manifolds with corner

This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities.

The mean curvature estimate for the level sets of solutions of the Monge-Ampère equation on Riemannian manifold

For the fully nonlinear elliptic Monge-Ampere equation det D 2 u = 1 with homogeneous Dirichlet boundary value condition, in this paper, a function related to the curvature of the level set of the solution was established, then the differential inequality of the strictly convex solutions of the equation on four-dimensional Riemannian manifold was got. The maximum value of the auxiliary function at the boundary was obtained by using the maximum principle and the mean curvature estimation for the level sets of the solution was given.

Four-dimensional almost Einstein manifolds with skew-circulant stuctures

We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.

Einstein Four-Manifolds with Sectional Curvature Bounded from Above

Given an Einstein structure $${\bar{g}}$$ with positive scalar curvature on a four-dimensional Riemannian manifold, that is $${\bar{R}}ic=\lambda {\bar{g}}$$ for some positive constant $$\lambda $$, a basic problem is to classify such Einstein 4-manifolds with positive or nonnegative sectional curvature. For convenience, the Ricci curvature is always normalized to be $${\bar{R}}ic=1$$. In this paper, we firstly show that if the sectional curvature of $${\bar{g}}$$ satisfies $${\bar{K}}\le \frac{\sqrt{3}}{2}\approx 0.866025$$, then $${\bar{g}}$$ must have nonnegative sectional curvature. Next, we prove a rigidity theorem of Einstein four-manifolds with nonnegative sectional curvature satisfying the additional condition that $${\bar{K}}_{ik}+s{\bar{K}}_{ij}\ge K_s$$ for every orthonormal basis $$\{e_i\}$$ with $${\bar{K}}_{ik}\ge {\bar{K}}_{ij}$$, where s is some nonnegative constant. More precisely, we show that such Einstein manifolds must be isometric to either $$S^4$$, or $$RP^4$$, or $$CP^2$$ (with standard metrics respectively). As a corollary, we obtain a rigidity result of Einstein four-manifolds with $${\bar{R}}ic=1$$ and the sectional curvature satisfying the upper bound $${\bar{K}} \le M_2 \approx 0.750912$$.

Minimal hyperspheres of arbitrarily large Morse index

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing Riemannian metrics on S^4 that admit embedded minimal hyperspheres of uniformly bounded volume and arbitrarily large Morse index. The phenomena we exhibit are in striking contrast with the three-dimensional compactness results by Choi-Schoen.

Levy Laplacian on a Four-Dimensional Riemannian Manifold

It is known that the Yang-Mills equations for a connection are equivalent to the Levy-Laplace equation for the parallel transport associated with this connection (see the paper [4] by L. Accardi, P. Gibilisco and I. V. Volovich). The Levy-Laplace equation is the Laplace equation for Levy Laplacian, which can be defined as the Cesaro mean of the second-order directional derivatives along the vectors from the orthonormal basis of some Hilbert space. The author’s work [11] has proved for the case of a four-dimensional Euclidean space, that with a certain choice of the orthonormal basis, the Laplace-Levy equation for a parallel transport becomes equivalent to the self-duality equations for a connection. A connection that is a solution to the self-duality equations is called an instanton and is a solution to the Yang-Mills equations. In the paper we define the Levy Laplacian for the case of a four-dimensional Riemannian manifold. This operator is a generalization of both the Levy Laplacian, introduced by the author in [11], and the Levy Laplacian, introduced in [3] by L. Accardi and О. G. Smolyanov for a Riemannian manifold. In the paper we consider the case of a line bundle over a four-dimensional Riemannian manifold and Maxwell's equations (the commutative case of the Yang-Mills equations). We find the conditions, which imply that from the fact that the parallel transport is a harmonic functional for the introduced Levy Laplacian follows that the appropriate connection is a solution to the self-duality equations. In addition, the paper considers the relationship of the introduced Levy Laplacian and the Laplace-Beltrami operator on a manifold. It can be expected that the results of the paper can be generalized to the non-commutative case of Yang-Mills fields.

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.

Test of emergent gravity

In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U(1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U(1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U(1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that the K\"ahler manifold determined by the Braden-Nekrasov instanton exhibits a spacetime singularity while the Nekrasov-Schwarz instanton gives rise to a regular geometry-the Eguchi-Hanson space. This result implies that the noncommutativity of spacetime plays an important role for the resolution of spacetime singularities in general relativity. We also discuss how the topological invariants associated with noncommutative U(1) instantons are related to those of emergent four-dimensional Riemannian manifolds according to the emergent gravity picture.

A note on supersymmetric AdS6 solutions of massive type IIA supergravity

Motivated by the AdS_6/CFT_5 correspondence, we study general supersymmetric solutions of massive type IIA supergravity, consisting of a warped product of six-dimensional anti-de Sitter space AdS_6 with a four-dimensional Riemannian manifold M_4, and fluxes compatible with the SO(5,2) symmetry. We find that the only local solution of this form is the warped AdS_6 x_w S^4 solution, originally discovered by Brandhuber and Oz. We discuss the supersymmetric properties of this solution.

Equivalence between the Osserman condition and the Rakić duality principle in dimension 4

We show that four-dimensional Riemannian manifolds which satisfy the Rakić duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant). Thus, since it was proved in Rakić (1999) [9] that Osserman manifolds satisfy the Rakić duality principle, both conditions are equivalent.

On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts, assuming an eigenvalue condition on the Cauchy-Riemann operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on the complex plane; (2) an existence theorem for holomorphic sections with controlled growth by Hormander's weighted L^2-method.

A uniform estimate for scalar curvature equation on manifolds of dimension 4

We consider the solutions to the prescribed scalar curvature equation on a four-dimensional Riemannian manifold M. We prove an upper bound for the supremum of all the solutions on every compact subset K of M, provided that all the solutions on M are bounded from below by a positive number.

H-CONTACT UNIT TANGENT SPHERE BUNDLES OF FOUR-DIMENSIONAL RIEMANNIAN MANIFOLDS

AbstractWe study the geometric properties of a base manifold whose unit tangent sphere bundle, equipped with the standard contact metric structure, is H-contact. We prove that a necessary and sufficient condition for the unit tangent sphere bundle of a four-dimensional Riemannian manifold to be H-contact is that the base manifold is 2-stein.

Integral pinching results for manifolds with boundary

We prove that some Riemannian manifolds with boundary satisfying an explicit integral pinching condition are spherical space forms. More precisely, we show that three-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L-norm of the scalar curvature and the L-norm of the Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that four-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of the (Q,T )-curvature and the L-norm of the Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence, we show that a certain conformally invariant operator, which plays an important role in Conformal Geometry, is non-negative and has trivial kernel if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the (Q,T )-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant Q-curvature, constant T -curvature, and zero mean curvature under the latter assumptions.

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.

Area–angle variables for general relativity

We introduce a modified Regge calculus for general relativity on a triangulated four-dimensional Riemannian manifold where the fundamental variables are areas and a certain class of angles. These variables satisfy constraints which are local in the triangulation. We expect the formulation to have applications to classical discrete gravity and non-perturbative approaches to quantum gravity.