Codazzi Tensor Research Articles

Codazzi tensors with two eigenvalue functions

This paper addresses a gap in the classification of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the eigenspaces is n − 1 n-1 , then the metric is a warped product where the base is an open interval, a conclusion we will show to be true under a milder trace condition. Furthermore, we construct examples of Codazzi tensors having two eigenvalue functions, one of which has eigenspace dimension n − 1 n-1 , where the metric is not a warped product with interval base, refuting a claim by A. L. Besse that the warped product conclusion holds without any restriction on the trace.

PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds ( PS )n and pseudo-concircular symmetric manifolds [Formula: see text] is defined. This is named pseudo-Q-symmetric and denoted with ( PQS )n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat ( PQS )n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of ( PQS )n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a ( PQS )n scalar field space-time is considered, and interesting properties are pointed out.

PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

Extended Derdziński–Shen theorem for curvature tensors

We extend a classical result by Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of some well known differential structures.

Riemann compatible tensors

Derdziński and Shen's theorem on the restrictions on the Riemann tensor imposed by existence of a Codazzi tensor holds more generally when a Riemann compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann co

DIRAC EIGENVALUES ESTIMATES IN TERMS OF DIVERGENCEFREE SYMMETRIC TENSORS

We proved in [10] that Friedrich's estimate [5] for the first eigenvalue of the Dirac operator can be improved when a Codazzi tensor exists. In the paper we further prove that his estimate can be improved as well via a well-chosen divergencefree symmetric tensor. We study the geometric implication of the new first eigenvalue estimates over Sasakian spin manifolds and show that some particular types of spinors appear as the limiting case.

EIGENVALUES ESTIMATES FOR THE DIRAC OPERATOR IN TERMS OF CODAZZI TENSORS

We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].

Hypersurfaces and Codazzi tensors

In this paper we deal with the following problem. Let (Mn,〈,〉) be an n-dimensional Riemannian manifold and \(f:(M^{n},\langle \,,\rangle)\rightarrow {\Bbb R}^{n+1}\) an isometric immersion. Find all Riemannian metrics on Mn that can be realized isometrically as immersed hypersurfaces in the Euclidean space \({\Bbb R}^{n+1}\). More precisely, given another Riemannian metric \(\widetilde{{\langle \,,\rangle }}\) on Mn, find necessary and sufficient conditions such that the Riemannian manifold \((M^{n},\widetilde{{\langle \,,\rangle}})\) admits an isometric immersion \({\tilde{f}}\) into the Euclidean space \({\Bbb R}^{n+1}\). If such an isometric immersion exists, how can one describe \({\tilde{f}}\) in terms of f?

Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise.

Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold

We study calibrated complex structures on the generalized tangent bundle of a Riemannian manifold M and their relationship to the Riemannian geometry of M. In particular we introduce a concept of integrability of such structures and we prove that integrability conditions are strictly related to the existence of certain Codazzi tensors on M.

Generalized cylinders in semi-Riemannian and spin geometry

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

A new variational characterization of 𝑛-dimensional space forms

A Riemannian manifold ( M n , g ) (M^n,g) is associated with a Schouten ( 0 , 2 ) (0,2) -tensor C g C_g which is a naturally defined Codazzi tensor in case ( M n , g ) (M^n,g) is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional F k [ g ] = ∫ M σ k ( C g ) d v o l g \mathcal {F}_k[g]=\int _M\sigma _k(C_g)dvol_g defined on M 1 = { g ∈ M | V o l ( g ) = 1 } \mathcal {M}_1=\{g\in \mathcal {M}|Vol(g)=1\} , where M \mathcal {M} is the space of smooth Riemannian metrics on a compact smooth manifold M M and { σ k ( C g ) , 1 ≤ k ≤ n } \{\sigma _k(C_g),\ 1\leq k\leq n\} is the elementary symmetric functions of the eigenvalues of C g C_g with respect to g g . We prove that if n ≥ 5 n\geq 5 and a conformally flat metric g g is a critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 , then g g must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky’s very recent theorem that the critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 characterized the three-dimensional space forms.

Commuting Codazzi tensors and the Ribaucour transformation for submanifolds

In this paper we develop a general theory for the Ribaucour transformation of sub-manifolds. In the process, a beautiful correspondence between Ribaucour transforms of a submanifold and Codazzi tensors that commute with its second fundamental form is revealed.

ON SEMI-RIEMANNIAN MANIFOLDS SATISFYING THE SECOND BIANCHI IDENTITY

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concir- cular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in (18) and (19).

The convex-concave principle and uniqueness for hypersurfaces in space Forms

We establish very general Weyl identities for pairs of symmetric functions of the invariants of the shape operator of a hypersurface in a space form or a general Codazzi tensor, respectively. They are used to characterize certain isoparametric hypersurfaces by assuming constancy or extremal properties of the functions, provided they fulfill ellipticity and/or convexity (concavity) properties. This way, many wellknown results are generalized. Finally, a chain rule for Weyl identities offers additional extension of some results.

Codazzi Tensors and the Topology of Surfaces

We introduce the notion of (0,m)-Codazzi tensors relative to an affine connection which extends the well known concept for m = 2. On compact Riemannian surfaces of genus γ we determine the dimension of the R-vector space of traceless Codazzi tensors, which depends on m and γ only, additionally we extend these result for genus zero. We give some exemplary applications to submanifolds in affine and in Riemannian geometry and to Weyl geometries.

Characterization of affine ruled surfaces

The aim of this paper is to give certain conditions characterizing ruled affine surfaces in terms of the Blaschke structure (∇, h, S) induced on a surface (M, f) in ℝ3. The investigation of affine ruled surfaces was started by W. Blaschke in the beginning of our century (see [1]). The description of affine ruled surfaces can be also found in the book [11], [3] and [7]. Ruled extremal surfaces are described in [9]. We show in the present paper that a shape operator S is a Codazzi tensor with respect to the Levi-Civita connection ∇ of affine metric h if and only if (M, f) is an affine sphere or a ruled surface. Affine surfaces with ∇S = 0 are described in [2] (see also [4]). We also show that a surface which is not an affine sphere is ruled iff im(S - HI) =ker(S - HI) and ket(S - HI) ⊂ ker dH. Finally we prove that an affine surface with indefinite affine metric is a ruled affine sphere if and only if the difference tensor K is a Codazzi tensor with respect to ∇.

CODAZZI TENSORS ON 2-DIMENSIONAL SPHERES

Ⅰ. INTRODUCTIONLet M be a Riemannian manifold with the Levi-Civita connection ▽. A symmetric 2-tensor field A on M is called a Codazzi tensor if the covariant derivative ▽A of A is a symmetric 3-tensor field. The notion, obviously, originates from the second fundamental

Harmonic Gauss maps and Codazzi tensors for affine hypersurfaces

Harmonic Gauss maps and Codazzi tensors for affine hypersurfaces

Some theorems on Codazzi tensors and their applications to hypersurfaces in Riemannian manifold of constant curvature

Some theorems on Codazzi tensors and their applications to hypersurfaces in Riemannian manifold of constant curvature