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Journal articles on the topic 'Codazzi tensor'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Ali, Mohabbat, Abdul Haseeb, Fatemah Mofarreh, and Mohd Vasiulla. "Z-Symmetric Manifolds Admitting Schouten Tensor." Mathematics 10, no. 22 (November 16, 2022): 4293. http://dx.doi.org/10.3390/math10224293.

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The paper deals with the study of Z-symmetric manifolds (ZS)n admitting certain cases of Schouten tensor (specifically: Ricci-recurrent, cyclic parallel, Codazzi type and covariantly constant), and investigate some geometric and physical properties of the manifold. Moreover, we also study (ZS)4 spacetimes admitting Codazzi type Schouten tensor. Finally, we construct an example of (ZS)4 to verify our result.

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2

Ünal, İnan. "Generalized Quasi-Einstein Manifolds in Contact Geometry." Mathematics 8, no. 9 (September 16, 2020): 1592. http://dx.doi.org/10.3390/math8091592.

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In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to M-projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold S2n+1(1)×S1.

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3

Stepanov, S., and I. Tsyganok. "Vanishing theorems for higher-order Killing and Codazzi." Differential Geometry of Manifolds of Figures, no. 50 (2019): 141–47. http://dx.doi.org/10.5922/0321-4796-2019-50-16.

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A Killing p-tensor (for an arbitrary natural number p ≥ 2) is a symmetric p-tensor with vanishing symmetrized covariant derivative. On the other hand, Codazzi p-tensor is a symmetric p-tensor with symmetric covariant derivative. Let M be a complete and simply connected Riemannian manifold of nonpositive (resp. non-negative) sectional curvature. In the first case we prove that an arbitrary symmetric traceless Killing p-tensor is parallel on M if its norm is a Lq -function for some q > 0. If in addition the volume of this manifold is infinite, then this tensor is equal to zero. In the second case we prove that an arbitrary traceless Codazzi p-tensor is equal to zero on a noncompact manifold M if its norm is a Lq -function for some q  1 .

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4

Calviño-Louzao, E., E. García-Río, J. Seoane-Bascoy, and R. Vázquez-Lorenzo. "Three-dimensional manifolds with special Cotton tensor." International Journal of Geometric Methods in Modern Physics 12, no. 01 (December 28, 2014): 1550005. http://dx.doi.org/10.1142/s021988781550005x.

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The Cotton tensor of three-dimensional Walker manifolds is investigated. A complete description of all locally conformally flat Walker three-manifolds is given, as well as that of Walker manifolds whose Cotton tensor is either a Codazzi or a Killing tensor.

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5

Mantica, Carlo Alberto, and Luca Guido Molinari. "Weyl compatible tensors." International Journal of Geometric Methods in Modern Physics 11, no. 08 (September 2014): 1450070. http://dx.doi.org/10.1142/s0219887814500704.

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We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdziński and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.

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6

MANTICA, CARLO ALBERTO, and YOUNG JIN SUH. "PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS." International Journal of Geometric Methods in Modern Physics 09, no. 01 (February 2012): 1250004. http://dx.doi.org/10.1142/s0219887812500041.

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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.

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7

Calvaruso, Giovanni. "Riemannian 3-metrics with a generic Codazzi Ricci tensor." Geometriae Dedicata 151, no. 1 (September 5, 2010): 259–67. http://dx.doi.org/10.1007/s10711-010-9532-5.

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8

Chen, Zhengmao. "A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure." Electronic Research Archive 31, no. 2 (2022): 840–59. http://dx.doi.org/10.3934/era.2023042.

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<abstract><p>In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.</p></abstract>

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9

Jelonek, Włodzimierz. "Characterization of affine ruled surfaces." Glasgow Mathematical Journal 39, no. 1 (January 1997): 17–20. http://dx.doi.org/10.1017/s0017089500031852.

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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 ∇.

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10

NAKAD, ROGER. "THE ENERGY-MOMENTUM TENSOR ON Spinc MANIFOLDS." International Journal of Geometric Methods in Modern Physics 08, no. 02 (March 2011): 345–65. http://dx.doi.org/10.1142/s0219887811005178.

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On Spinc manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spinc manifold. Using the notion of generalized cylinders, we derive the variational formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spinc Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.

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11

Suh, Young Jin, Carlo Alberto Mantica, Uday Chand De, and Prajjwal Pal. "Pseudo B-symmetric manifolds." International Journal of Geometric Methods in Modern Physics 14, no. 09 (August 2, 2017): 1750119. http://dx.doi.org/10.1142/s0219887817501195.

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In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

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12

ROTH, JULIEN. "A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS." Bulletin of the Australian Mathematical Society 91, no. 1 (October 14, 2014): 145–54. http://dx.doi.org/10.1017/s0004972714000732.

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AbstractIn this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms.

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13

Reshetnyak, Yuri G. "On the Stability in Bonnet's Theorem of the Surface Theory." gmj 14, no. 3 (September 2007): 543–64. http://dx.doi.org/10.1515/gmj.2007.543.

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Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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14

Mallick, Sahanous, and Uday Chand De. "Spacetimes admitting W2-curvature tensor." International Journal of Geometric Methods in Modern Physics 11, no. 04 (April 2014): 1450030. http://dx.doi.org/10.1142/s0219887814500303.

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The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

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15

De, Krishnendu, and Uday Chand De. "η-Ricci Solitons on Kenmotsu 3-Manifolds." Annals of West University of Timisoara - Mathematics and Computer Science 56, no. 1 (July 1, 2018): 51–63. http://dx.doi.org/10.2478/awutm-2018-0004.

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Abstract In the present paper we study η-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider η-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study φ-Ricci symmetric η-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition R.R = Q(S, R)is considered. Finally, an example is constructed to prove the existence of a proper η-Ricci soliton on a Kenmotsu 3-manifold.

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16

Wang, Wenjie, and Ximin Liu. "Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions." Filomat 32, no. 1 (2018): 197–206. http://dx.doi.org/10.2298/fil1801197w.

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In this paper, a three-dimensional almost Kenmotsu manifold M3 satisfying the generalized (k,?)'-nullity condition is investigated. We mainly prove that on M3 the following statements are equivalent: (1) M3 is ?-symmetric; (2) the Ricci tensor of M3 is cyclic-parallel; (3) the Ricci tensor of M3 is of Codazzi type; (4) M3 is conformally flat with scalar curvature invariant along the Reeb vector field; (5) M3 is locally isometric to either the hyperbolic space H3(-1) or the Riemannian product H2(-4) x R.

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17

Yasar, Erol, A. Ceylan Cöken, and Ahmet Yücesan. "Lightlike hypersurfaces in semi-Riemannian manifold with semi-symmetric non-metric connection." MATHEMATICA SCANDINAVICA 102, no. 2 (June 1, 2008): 253. http://dx.doi.org/10.7146/math.scand.a-15061.

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In this paper, we study lightlike hypersurfaces of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. We give the equations of Gauss and Codazzi. Then, we obtain conditions under which the Ricci tensor of a lightlike hypersurface is symmetric given that the ambient space is equipped with a semi-symmetric non-metric connection.

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18

MANTICA, CARLO ALBERTO, and YOUNG JIN SUH. "PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS." International Journal of Geometric Methods in Modern Physics 10, no. 05 (April 3, 2013): 1350013. http://dx.doi.org/10.1142/s0219887813500138.

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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.

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Majhi, Pradip, Uday Chand De, and Debabrata Kar. "η-Ricci Solitons on Sasakian 3-Manifolds." Annals of West University of Timisoara - Mathematics and Computer Science 55, no. 2 (December 1, 2017): 143–56. http://dx.doi.org/10.1515/awutm-2017-0019.

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AbstractIn this paper we studyη-Ricci solitons on Sasakian 3-manifolds. Among others we prove that anη-Ricci soliton on a Sasakian 3-manifold is anη-Einstien manifold. Moreover we considerη-Ricci solitons on Sasakian 3-manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Beside these we study conformally flat andφ-Ricci symmetricη-Ricci soliton on Sasakian 3-manifolds. Alsoη-Ricci soliton on Sasakian 3-manifolds with the curvature conditionQ.R= 0 have been considered. Finally, we construct an example to prove the non-existence of properη-Ricci solitons on Sasakian 3-manifolds and verify some results.

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20

Mincic, Svetislav, Ljubica Velimirovic, and Mica Stankovic. "New integrability conditions of derivational equations of a submanifold in a generalized Riemannian space." Filomat 24, no. 4 (2010): 137–46. http://dx.doi.org/10.2298/fil1004137m.

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The present work is a continuation of [5] and [6]. In [5] we have obtained derivational equations of a submanifold XM of a generalized Riemannian space GRN. Since the basic tensor in GRN is asymmetric and in this way the connection is also asymmetric, in a submanifold the connection is generally asymmetric too. By reason of this, we define 4 kinds of covariant derivative and obtain 4 kinds of derivational equations. In [6] we have obtained integrability conditions and Gauss-Codazzi equations using the 1st and the 2st kind of covariant derivative. The present work deals in the cited matter, using the 3rd and the 4th kind of covariant derivative. One obtains three new integrability conditions for derivational equations of tangents and three such conditions for normals of the submanifold, as the corresponding Gauss-Codazzi equations too.

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SHAIKH, ABSOS ALI, TRAN QUOC BINH, and HARADHAN KUNDU. "Curvature Properties of Generalized pp-Wave Metrics." Kragujevac Journal of Mathematics 45, no. 02 (April 2021): 237–58. http://dx.doi.org/10.46793/kgjmat2102.237s.

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The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metrics. It is shown that a generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisﬁes the pseudosymmetric type condition P ⋅ P = −13Q(S,P). Again we investigate the suﬃcient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent. Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. Again the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally, we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.

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22

YANG, GUO-HONG, SHI-XIANG FENG, GUANG-JIONG NI, and YI-SHI DUAN. "RELATIONS OF TWO TRANSVERSAL SUBMANIFOLDS AND GLOBAL MANIFOLD." International Journal of Modern Physics A 16, no. 21 (August 20, 2001): 3535–51. http://dx.doi.org/10.1142/s0217751x01005080.

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In Riemann geometry, the relations of two transversal submanifolds and global manifold are discussed without any concrete models. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are connected at the intersection points of the two submanifolds. When the inner product of the two tangent vectors of submanifolds vanishes, some corollaries of these relations give the most important second fundamental form and Gauss–Codazzi equation in the conventional submanifold theory. As a special case, the global manifold which is Euclidean is considered. It is pointed out that, in order to obtain the nonzero energy–momentum tensor of matter field in a submanifold, there must be the contributions of the above inner product and the other submanifold. Generally speaking, a submanifold is closely related to the matter fields of the other submanifold and the two submanifolds affect each other through the above inner product.

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23

De, Uday Chand, Sameh Shenawy, Abdallah Abdelhameed Syied, and Nasser Bin Turki. "Conformally Flat Pseudoprojective Symmetric Spacetimes in f R , G Gravity." Advances in Mathematical Physics 2022 (March 25, 2022): 1–7. http://dx.doi.org/10.1155/2022/3096782.

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Sufficient conditions on a pseudoprojective symmetric spacetime PPS n whose Ricci tensor is of Codazzi type to be either a perfect fluid or Einstein spacetime are given. Also, it is shown that a PPS n is Einstein if its Ricci tensor is cyclic parallel. Next, we illustrate that a conformally flat PPS n spacetime is of constant curvature. Finally, we investigate conformally flat PPS 4 spacetimes and conformally flat PPS 4 perfect fluids in f R , G theory of gravity, and amongst many results, it is proved that the isotropic pressure and the energy density of conformally flat perfect fluid PPS 4 spacetimes are constants and such perfect fluid behaves like a cosmological constant. Further, in this setting, we consider some energy conditions.

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Pande, H. D., and A. Kumar. "Generalisation of Gauss--Codazzi equations for Berwald's curvature tensor in a hypersurface of a Finsler space." Publicationes Mathematicae Debrecen 22, no. 3-4 (July 1, 2022): 263–67. http://dx.doi.org/10.5486/pmd.1975.22.3-4.13.

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Suh, Young, and Uday De. "On a type of spacetimes." Filomat 33, no. 13 (2019): 4251–60. http://dx.doi.org/10.2298/fil1913251s.

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In the present paper we characterize a type of spacetimes, called almost pseudo Z-symmetric spacetimes A(PZS)4. At first, we obtain a condition for an A(PZS)4 spacetime to be a perfect fluid spacetime and Roberson-Walker spacetime. It is shown that an A(PZS)4 spacetime is a perfect fluid spacetime if the Z tensor is of Codazzi type. Next we prove that such a spacetime is the Roberson-Walker spacetime and can be identified with Petrov types I, D or O[3], provided the associated scalar ? is constant. Then we investigate A(PZS)4 spacetimes satisfying divC = 0 and state equation is derived. Also some physical consequences are outlined. Finally, we construct a metric example of an A(PZS)4 spacetime.

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Baḡdatlı Yılmaz, Hülya, and S. Aynur Uysal. "Compatibility of φ(Ric)-vector fields on almost pseudo-Ricci symmetric manifolds." International Journal of Geometric Methods in Modern Physics 18, no. 08 (May 12, 2021): 2150128. http://dx.doi.org/10.1142/s0219887821501280.

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The object of the paper is to study the compatibility of [Formula: see text]-vector fields on almost pseudo-Ricci symmetric manifolds, briefly [Formula: see text]. First, we show the existence of an [Formula: see text] whose basic vector field [Formula: see text] is a [Formula: see text]-vector field by constructing a non-trivial example. Then, we investigate the properties of the Riemann and Weyl compatibility of [Formula: see text] under certain conditions. We consider an [Formula: see text] space-time whose basic vector fields [Formula: see text] and [Formula: see text] is [Formula: see text]-vector fields of constant length. Moreover, we show that an [Formula: see text] space-time whose Ricci tensor is of Codazzi type and basic vector field [Formula: see text] is [Formula: see text]-vector field is purely electric space-time.

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Mantica, Carlo Alberto, and Young Jin Suh. "Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves." International Journal of Geometric Methods in Modern Physics 13, no. 02 (January 26, 2016): 1650015. http://dx.doi.org/10.1142/s0219887816500158.

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In this paper we present some new results about [Formula: see text]-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for [Formula: see text] and a pp-wave space-time in [Formula: see text]. In all cases an algebraic classification for the Weyl tensor is provided for [Formula: see text] and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat [Formula: see text], [Formula: see text], space-time is conformal to the Robertson–Walker space-time.

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CIARLET, PHILIPPE G., LILIANA GRATIE, CRISTINEL MARDARE, and MING SHEN. "SAINT VENANT COMPATIBILITY EQUATIONS ON A SURFACE APPLICATION TO INTRINSIC SHELL THEORY." Mathematical Models and Methods in Applied Sciences 18, no. 02 (February 2008): 165–94. http://dx.doi.org/10.1142/s0218202508002644.

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We first establish that the linearized change of metric and change of curvature tensors, with components in L2 and H-1 respectively, associated with a displacement field, with components in H1, of a surface S immersed in ℝ3 must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient, i.e. that they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ ℝ3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established. The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. This algorithm may be viewed as the linear counterpart of the reconstruction of a surface from its first and second fundamental forms. Finally, we show how these results can be applied to the "intrinsic theory" of linearly elastic shells, where the linearized change of metric and change of curvature tensors are the new unknowns. These new unknowns solve a quadratic minimization problem over a space of tensor fields whose components, which are only in L2, satisfy the Saint Venant compatibility conditions on a surface in the sense of distributions.

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29

Miao, Jiajing, Jinli Yang, and Jianyun Guan. "Classification of Lorentzian Lie Groups Based on Codazzi Tensors Associated with Yano Connections." Symmetry 14, no. 8 (August 18, 2022): 1730. http://dx.doi.org/10.3390/sym14081730.

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In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with Yano connections are Codazzi tensors. The main results are listed in a table, and indicate that G1 and G7 do not have Codazzi tensors associated with Yano connections, G2, G3, G4, G5 and G6 have Codazzi tensors associated with Yano connections.

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30

Hasanis, Thomas, and Theodoros Vlachos. "Hypersurfaces and Codazzi tensors." Monatshefte für Mathematik 154, no. 1 (March 3, 2008): 51–58. http://dx.doi.org/10.1007/s00605-008-0528-2.

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31

An-Min, Li. "Some theorems on Codazzi tensors." Mathematische Zeitschrift 191, no. 4 (December 1986): 575–84. http://dx.doi.org/10.1007/bf01162347.

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32

Catino, Giovanni, Carlo Mantegazza, and Lorenzo Mazzieri. "A note on Codazzi tensors." Mathematische Annalen 362, no. 1-2 (November 22, 2014): 629–38. http://dx.doi.org/10.1007/s00208-014-1135-2.

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33

Merton, Gabe. "Codazzi tensors with two eigenvalue functions." Proceedings of the American Mathematical Society 141, no. 9 (May 16, 2013): 3265–73. http://dx.doi.org/10.1090/s0002-9939-2013-11616-3.

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34

Wu, Tong, and Yong Wang. "Codazzi Tensors and the Quasi-Statistical Structure Associated with Affine Connections on Three-Dimensional Lorentzian Lie Groups." Symmetry 13, no. 8 (August 9, 2021): 1459. http://dx.doi.org/10.3390/sym13081459.

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In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with these connections. We also classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with Bott connections, canonical connections and Kobayashi–Nomizu connections.

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35

Stepanov, S. E., and I. I. Tsyganok. "Codazzi and Killing Tensors on a Complete Riemannian Manifold." Mathematical Notes 109, no. 5-6 (May 2021): 932–39. http://dx.doi.org/10.1134/s0001434621050266.

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36

Dajczer, Marcos, and Ruy Tojeiro. "Commuting Codazzi tensors and the Ribaucour transformation for submanifolds." Results in Mathematics 44, no. 3-4 (November 2003): 258–78. http://dx.doi.org/10.1007/bf03322986.

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37

Shen, Yi Bing. "Harmonic Gauss maps and Codazzi tensors for affine hypersurfaces." Archiv der Mathematik 55, no. 3 (September 1990): 298–305. http://dx.doi.org/10.1007/bf01191173.

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38

Shandra, Igor G., Sergey E. Stepanov, and Josef Mikeš. "On higher-order Codazzi tensors on complete Riemannian manifolds." Annals of Global Analysis and Geometry 56, no. 3 (July 6, 2019): 429–42. http://dx.doi.org/10.1007/s10455-019-09673-w.

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39

Stepanov, S. E., I. I. Tsyganok, and J. Mikeš. "Complete Riemannian manifolds with Killing — Ricci and Codazzi — Ricci tensors." Differential Geometry of Manifolds of Figures, no. 53 (2022): 112–17. http://dx.doi.org/10.5922/0321-4796-2022-53-10.

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The purpose of this paper is to prove of Liouville type theorems, i. e., theorems on the non-existence of Killing — Ricci and Codazzi — Ricci tensors on complete non-compact Riemannian manifolds. Our results complement the two classical vanishing theorems from the last chapter of famous Besse’s monograph on Einstein manifolds.

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40

Friedrich, Thomas, and Eui-Chul Kim. "EIGENVALUES ESTIMATES FOR THE DIRAC OPERATOR IN TERMS OF CODAZZI TENSORS." Bulletin of the Korean Mathematical Society 45, no. 2 (May 31, 2008): 365–73. http://dx.doi.org/10.4134/bkms.2008.45.2.365.

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41

Ganguly, D., S. Dey, and A. Bhattacharyya. "On trans-Sasakian $3$-manifolds as $\eta$-Einstein solitons." Carpathian Mathematical Publications 13, no. 2 (October 15, 2021): 460–74. http://dx.doi.org/10.15330/cmp.13.2.460-474.

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The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $\eta$-Einstein soliton to verify our results.

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42

SZOPOS, MARCELA. "ON THE RECOVERY AND CONTINUITY OF A SUBMANIFOLD WITH BOUNDARY." Analysis and Applications 03, no. 02 (April 2005): 119–43. http://dx.doi.org/10.1142/s0219530505000510.

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The fundamental theorem of Riemannian geometry asserts that a connected and simply-connected Riemannian space ω of ℝp can be isometrically immersed into the Euclidean space ℝp+q if and only if there exist tensors satisfying the Gauss–Ricci–Codazzi equations, in which case these immersions are uniquely determined up to isometries in ℝp+q. In this fashion, we can define a mapping which associates with these prescribed tensors the reconstructed submanifold. The purpose of this paper is twofold: under a smoothness assumption on the boundary of ω, we first establish an analogous result for the existence and uniqueness of a submanifold "with boundary" and then show that the mapping constructed in this fashion is locally Lipschitz-continuous with respect to the topology of the Banach spaces [Formula: see text].

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43

Siddiqi, Mohd Danish, Fatemah Mofarreh, Aliya Naaz Siddiqui, and Shah Alam Siddiqui. "Geometrical Structure in a Relativistic Thermodynamical Fluid Spacetime." Axioms 12, no. 2 (January 29, 2023): 138. http://dx.doi.org/10.3390/axioms12020138.

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The goal of the present research paper is to study how a spacetime manifold evolves when thermal flux, thermal energy density and thermal stress are involved; such spacetime is called a thermodynamical fluid spacetime (TFS). We deal with some geometrical characteristics of TFS and obtain the value of cosmological constant Λ. The next step is to demonstrate that a relativistic TFS is a generalized Ricci recurrent TFS. Moreover, we use TFS with thermodynamic matter tensors of Codazzi type and Ricci cyclic type. In addition, we discover the solitonic significance of TFS in terms of the Ricci metric (i.e., Ricci soliton RS).

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(Nikolay) Ivanovich, Yaremenko Mikola. "Theory of Hypersurfaces yn–1 in yn Space and Geodesics on this Hypersurfaces yn–1." Asian Research Journal of Mathematics, March 28, 2019, 1–14. http://dx.doi.org/10.9734/arjom/2019/v13i230105.

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The hypersurface yn–1 in yn space is studied for this piece of work. We established the correlation between tensors of hypersurface yn–1 and tensors of embedding space yn . The second non-symmetrical tensor of hypersurface has been introduced, which have been obtained from the analog of Peterson-Codazzi equation in nonsymmetrical case.Also we have introduced the tensor that is associated with square of angle between normal and adjacent normal and it is represented in terms of metric and second tensors of hypersurface. The geodesics on hypersurface have been studied, and nontrivial example of geodesics on hypersurface with torsion and Euclid metric was constructed.

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45

Makhal, Sourav. "SOME RESULTS ON GENERALIZED $(k,\mu)$-PARACONTACT METRIC MANIFOLDS." Facta Universitatis, Series: Mathematics and Informatics, October 19, 2018, 401. http://dx.doi.org/10.22190/fumi1803401m.

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The object of this paper is to study Codazzi type of Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Next we study cyclic parallel Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Further, we characterized generalized $(k,\mu )$-paracontact metric manifolds whose structure tensor $\phi$ is $\eta$-parallel. Finally, we investigate locally $\phi$-Ricci symmetric generalized $(k,\mu )$-paracontact metric manifolds.

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46

TURANLI, Sibel, and Sedanur UÇAN. "Interaction of Codazzi Pairs with Almost Para Norden Manifolds." Turkish Journal of Mathematics and Computer Science, June 8, 2022. http://dx.doi.org/10.47000/tjmcs.1075806.

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In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an almost para Norden manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate connection of a linear connection $\mathrm{\nabla }$ on $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\ },\ {\mathrm{\nabla }}^{\dagger \ }$ and ${\mathrm{\nabla }}^{\varphi \ }$ are defined and it is demonstrated that on the spaces of linear connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as the four-element Klein group. We also searched some properties of these three types conjugate connections. Then, Codazzi pairs $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ and $\left(\mathrm{\nabla },G\right)$ are introduced and some properties of them are given. Let $R\ ,\ R^{*\ }$and $R^{\dagger \ }$are $(0,4)$-curvature tensors of conjugate connections $\mathrm{\nabla }\mathrm{\ ,\ }{\mathrm{\nabla }}^{*\ }$and ${\mathrm{\nabla }}^{\dagger \ }$, respectively. The relationship among the curvature tensors is investigated. The condition of $N_{\varphi }=0$ is obtained, where $N_{\varphi }$ is Nijenhuis tensor field on $M_{2n}$ and it is known that the condition of integrability of almost para complex structure $\varphi $ is $N_{\varphi }=0$. In addition, Tachibana operator is applied to the pure metric $g$ and a necessary and sufficient condition $\left(M,\varphi ,\ g,G\right)$ being a para Kahler Norden manifold is found. Finally, we examine $\varphi $-invariant linear connections and statistical manifolds.

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47

Pandey, Shashikant, Abhishek Singh, and Vishnu Narayan Mishra. "ETA-RICCI SOLITONS ON LORENTZIAN PARA-KENMOTSU MANIFOLDS." Facta Universitatis, Series: Mathematics and Informatics, July 30, 2021, 419. http://dx.doi.org/10.22190/fumi200923031p.

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The objective of present research article is to investigate the geometric properties of $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds. In this manner, we consider $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds satisfying $R\cdot S=0$. Further, we obtain results for $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds with quasi-conformally flat property. Moreover, we get results for $\eta$-Ricci solitons in Lorentzian para-Kenmotsu manifolds admitting Codazzi type of Ricci tensor and cyclic parallel Ricci tensor, $\eta$-quasi-conformally semi-symmetric, $\eta$-Ricci symmetric and quasi-conformally Ricci semi-symmetric. At last, we construct an example of a such manifold which justify the existence of proper $\eta$-Ricci solitons.

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48

Li, Yanlin, Somnath Mondal, Santu Dey, Arindam Bhattacharyya, and Akram Ali. "A Study of Conformal $$\eta$$-Einstein Solitons on Trans-Sasakian 3-Manifold." Journal of Nonlinear Mathematical Physics, October 27, 2022. http://dx.doi.org/10.1007/s44198-022-00088-z.

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AbstractWe study conformal $$\eta$$ η -Einstein solitons on the framework of trans-Sasakian manifold in dimension three. Existence of conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold is discussed. Then we find some results on trans-Sasakian manifold which are conformal $$\eta$$ η -Einstein solitons where the Ricci tensor is cyclic parallel and Codazzi type. We also consider some curvature conditions with addition to conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold. We also use torse-forming vector fields in addition to conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold. Finally, an example of conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold is constructed.

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49

Matsumoto, Koji. "Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II." Proceedings of the International Geometry Center 11, no. 3 (January 21, 2019). http://dx.doi.org/10.15673/tmgc.v11i3.1202.

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In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}. In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}. Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form. In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold. Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}. In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form. Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}. Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$ satisfy some special equations, see Theorem~\ref{th5.2}.

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50

Ge̖barowski, Andrzej. "THE STRUCTURE OF CERTAIN RIEMANNIAN MANIFOLDS ADMITTING CODAZZI TENSORS." Demonstratio Mathematica 27, no. 1 (January 1, 1994). http://dx.doi.org/10.1515/dema-1994-0128.

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