Neutral Metric Research Articles

Null hypersurfaces in 4-manifolds endowed with a product structure

AbstractIn a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

We investigate minimal surfaces in products of two-spheres $\mathbf{S}^2_p\times\mathbf{S}^2_p$, with the neutral metric given by $(g,-g)$. Here $\mathbf{S}^2_p\subset \mathbf{R}^{p,3-p}$, and $g$ is the induced metric on the sphere. We compute all totally geodesic surfaces and we give a relation between minimal surfaces and the solutions of the Gordon equations. Finally, in some cases we give a topological classification of compact minimal surfaces.

Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

Horizontality in the twistor spaces associated with vector bundles of rank 4 on tori

Let E be an oriented vector bundle of rank 4 over a torus $$T^2$$ with a metric h and $$\hat{E}$$ the twistor space associated with E. We will study the horizontality in $$\hat{E}$$ with respect to the connection $$\hat{\nabla }$$ induced by an h-connection $$\nabla $$ . We will describe the subset of the fiber $$\hat{E}_a$$ of $$\hat{E}$$ at a point $$a\in T^2$$ given by horizontality along normal polygonal curves in $${{\varvec{R}}}^2$$ for an initial value at a in the case where the subset is finite. We will see that if $$\hat{E}$$ has a partially horizontal section $$\Omega $$ , then there exists an h-connection $$\nabla '$$ related to $$\nabla $$ such that h, $$\nabla '$$ and $$\Omega $$ give a Kahler structure of E. We will make analogous discussions for an oriented vector bundle of rank 4 over $$T^2$$ with a neutral metric.

Minimal Lorentz surfaces in pseudo-Euclidean 4-space with neutral metric

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature ϰ satisfy the inequality K2−ϰ2>0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.

Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature.

General Rotational Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classification of minimal general rotational surfaces of elliptic and hyperbolic type, general rotational surfaces with parallel normalized mean curvature vector field, flat general rotational surfaces, and general rotational surfaces with flat normal connection.

Lagrangian submanifolds in para-complex Euclidean space

We address the study of some curvature equations for distinguished submanifolds in para-Kahler geometry. We first observe that a para-complex submanifold of a para-Kahler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space $\mathbb{D}^n$ and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of $\mathbb{D}^2$ which are minimal and have indefinite metric. Finally we describe those Lagrangian self-similar solutions of the Mean Curvature Flow (with respect to the neutral metric of $\mathbb{D}^n$) which are $SO(n)$-equivariant.

Quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map in pseudo-Euclidean 4-space

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.

Dirac Structures on Banach Lie Algebroids

Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

In this paper we consider 4-dimensional neutral-signature curvature models and obtain the complete classification of the Ricci operator. We then consider the property of curvature homogeneity for the above manifolds and prove that every complete, connected and simply connected 1-curvature homogeneous 4-dimensional manifold of signature (2,2) with a non-degenerate Ricci operator is isometric to a four-dimensional Lie group equipped with a left invariant neutral metric. We also classify Ricci-parallel curvature homogeneous 4-dimensional manifolds of signature (2,2).

Compact Osserman Manifolds with Neutral Metric

It is shown that if a compact four-dimensional manifold with metric of neutral signature is Jordan–Osserman, then it is either of constant sectional curvature or Ricci flat.

Global smooth fibrations of ℝ3 by oriented lines

A smooth fibration of ℝ 3 by oriented lines is given by a smooth unit vector field V on ℝ 3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let ℒ be the space of oriented lines of ℝ 3 endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of ℝ 3 by oriented lines as the closed (in the relative topology) definite connected surfaces in ℒ. In particular, local conditions on ℒ imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle T G of a 2n-dimensional Lie group G, which are left invariant with respect to the Lie group structure on T G induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on G. Using this correspondence and results of (8) and (10), it turns out that when G is nilpotent and four or six dimensional, the cotangent bundle T G always has a hermitian structure. However, we prove that if G is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then T G has no hermitian structure or, equivalently, G has no left invariant generalized complex structure.

Classification of the Weyl curvature spinors of neutral metrics in four dimensions

Neutral geometry is of increasing interest. As with Riemannian and Lorentzian geometry, spinors can be expected to provide a valuable tool in neutral geometry. For a neutral metric in four dimensions, the classification of the Weyl curvature spinors by the pattern of principal spinors each admits is given. For each Weyl curvature spinor, there are nine nontrivial types. This classification is then related to the classification, given previously by the author, of a Weyl curvature spinor when regarded as a curvature endomorphism (four types). These results are the neutral analogues of well known and fundamental results in Lorentzian geometry, but display the peculiarities of neutral geometry. One can expect these results to be an essential ingredient in a full understanding of neutral geometry in four dimensions.

ParaHermitian and paraquaternionic manifolds

A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov–Gauduchon generalization of the Goldberg–Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira–Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on Sol 1 4 . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira–Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on Sol 1 4 is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed.

Isotropic Kähler hyperbolic twistor spaces

In this paper we study two natural indefinite almost Hermitian structures on the hyperbolic twistor space of a four-manifold endowed with a neutral metric. We show that only one of these structures can be isotropic Kähler and obtain the precise geometric conditions on the base manifold ensuring this property.

The holonomy Lie algebras of neutral metrics in dimension four

The Lie algebra isomorphism between su(1,1)×su(1,1) and o(2,2) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously known results, furnish a classification of such Lie subalgebras of o(2,2), with only one case remaining unresolved.