summary:We introduce a type of Riemannian manifolds (namely, quasirecurrent manifold) and study its several geometric properties. Among others, we prove that the scalar curvature of such a manifold is constant, and that the manifold is Einstein under certain condition. In addition, we deal with a quasirecurrent product manifold. Finally, we ensure the existence of quasirecurrent manifold by a proper example

Kim, Jaeman

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Institute of Mathematics AS CR, v. v. i.

Mathematica BohemicaJaeman KimOn quasirecurrent manifoldsMathematica Bohemica, Vol. 150 (2025), No. 3, 405–413Persistent URL: http://dml.cz/dmlcz/153084Terms of use:© Institute of Mathematics AS CR, 2025Institute of Mathematics of the Czech Academy of Sciences provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.cz150 (2025) MATHEMATICA BOHEMICA No. 3, 405–413ON QUASIRECURRENT MANIFOLDSJaeman KimReceived April 1, 2024. Published online December 6, 2024.Communicated by Pavel PyrihAbstract. We introduce a type of Riemannian manifolds (namely, quasirecurrent mani-fold) and study its several geometric properties. Among others, we prove that the scalarcurvature of such a manifold is constant, and that the manifold is Einstein under certaincondition. In addition, we deal with a quasirecurrent product manifold. Finally, we ensurethe existence of quasirecurrent manifold by a proper example.Keywords: quasirecurrent manifold; associated vector field; constant scalar curvature;Ricci symmetry; Einstein; cyclic Ricci symmetry; conformally flat; quasirecurrent productmanifold; space of constant curvatureMSC 2020 : 53A55, 53B201. IntroductionCartan [3] introduced the notion of (locally) symmetric manifolds as a general-ization of the notion of a space of constant curvature and he obtained a classifica-tion of such manifolds. The study on generalization of locally symmetric manifoldsstarted in 1946 and continued to date in different directions. For instance the no-tions of recurrent manifold and conformally recurrent manifold were introduced byRuse [8], [9], [10] and Walker [11]; Adati and Miyazawa [1], respectively. A Rie-mannian manifold (Mn, g) is called a recurrent manifold provided that its curvaturetensor R satisfies the relation(1.1) (∇XR)(Y, Z, U, V ) = A(X)R(Y, Z, U, V ),where ∇ denotes the Levi-Civita connection and A is a nonzero 1-form.Also a Riemannian manifold (Mn, g) is called a conformally recurrent manifoldprovided that its conformal curvature tensor C satisfies the relation(1.2) (∇XC)(Y, Z, U, V ) = A(X)C(Y, Z, U, V ),where A is a nonzero 1-form.c© The author(s) 2024. This is an open access article under the CC BY-NC-ND licence cbndDOI: 10.21136/MB.2024.0038-24 405The author has recently studied a type of weakly symmetric structure on a Rie-mannian manifold [4]. In [6], Pokhariyal defined some curvature tensors with thehelp of Weyl’s projective curvature tensor and studied their physical and geometricproperties. One of the curvature tensors introduced in [6] was the W2-curvaturetensor defined by(1.3) W2(X,Y, Z, U) = R(X,Y, Z, U) +1n− 1(g(X,Z)Ric(Y, U)− g(Y, Z)Ric(X,U)),where Ric denotes the Ricci tensor.In [5], [7], Pokhariyal et al. introduced the notion of W2-recurrent manifold andstudied its geometric properties. A Riemannian manifold (Mn, g) is said to be W2-recurrent provided that its W2-curvature tensor satisfies the relation(1.4) (∇XW2)(Y, Z, U, V ) = A(X)W2(Y, Z, U, V ),where A is a nonzero 1-form.Motivated by the above studies, as a sequel to [4], we introduce a type of Rie-mannian manifold called quasirecurrent manifold and study its several geometricproperties.A Riemannian manifold (Mn, g) is called a quasirecurrent manifold provided thatits curvature tensor R satisfies the relation(1.5) (∇XR)(Y, Z, U, V ) = A(X)W2(Y, Z, U, V ),where A is a nonzero 1-form.A quasirecurrent manifold with covariantly constant Ricci tensor is W2-recurrent,while a quasirecurrent manifold with vanishing Ricci tensor reduces to a recurrentmanifold. Hence, it is worthwhile to undertake the study of a quasirecurrent mani-fold. This paper is organized as follows. In Section 2, we give a sufficient conditionfor a quasirecurrent manifold to be Einstein. Section 3 is concerned with quasirecur-rent product manifold. In addition, a proper example of quasirecurrent manifold isprovided.2. Some properties of quasirecurrent manifoldLet (Mn, g) be a quasirecurrent manifold. Contracting (1.3) with respect to Xand U , we get(2.1) W2(Y, Z) =nn− 1(Ric(Y, Z)−sng(Y, Z)),where s denotes the scalar curvature.406Contracting (1.5) with respect to Y and V , from (2.1) we have(2.2) (∇XRic)(Z,U) = A(X)nn− 1(Ric(Z,U)−sng(Z,U)).Now we have the following theorem.Theorem 2.1. Let (Mn, g) be a quasirecurrent manifold of dimension n. Thenits scalar curvature s is constant.P r o o f. Contracting (2.2) with respect to Z and U , we getXs = 0,showing that the scalar curvature s of a quasirecurrent manifold is constant. A Riemannian manifold (Mn, g) is said to be Ricci-symmetric provided that itsRicci tensor is covariantly constant. In particular, a Riemannian manifold (Mn, g)is called an Einstein manifold provided that its Ricci tensor is proportional to themetric tensor, i.e., Ric = sg/n. In this case, its scalar curvature s is constant [2].Theorem 2.2. Let (Mn, g) be a quasirecurrent manifold which is Ricci-symme-tric. Then the manifold is Einstein.P r o o f. By taking account of (2.2) and A 6= 0, we know that the manifold isEinstein. Now we verify some relationships among recurrent manifold, quasirecurrent mani-fold, W2-recurrent manifold and conformally recurrent manifold.Theorem 2.3. Let (Mn, g) be a Riemannian manifold with covariantly constantRicci tensor. Then condition (1.5) of quasirecurrent manifold is equivalent to condi-tion (1.4) of W2-recurrent manifold. In particular, if a Riemannian manifold (Mn, g)is Einstein, then condition (1.5) of quasirecurrent manifold is equivalent to condi-tion (1.2) of conformally recurrent manifold. Furthermore, if a Riemannian manifold(Mn, g) has vanishing Ricci tensor, then condition (1.5) of quasirecurrent manifoldis equivalent to condition (1.1) of recurrent manifold.P r o o f. By taking account of (1.3), we have from ∇Ric = 0 the equivalencebetween relations (1.4) and (1.5). In particular, if the manifold is Einstein, it is wellknown [2] thatC(X,Y, Z, U) = R(X,Y, Z, U)−sn(n− 1)(g(X,U)g(Y, Z)− g(X,Z)g(Y, U)),407which yields(2.3) C(X,Y, Z, U) = W2(X,Y, Z, U)and(2.4) (∇XC)(Y, Z, U, V ) = (∇XR)(Y, Z, U, V )because of (1.3) and Ric = sg/n (and hence s = constant). Therefore by takingaccount of (2.3) and (2.4), we conclude that relation (1.5) holds if and only if relation(1.2) holds. Furthermore, considering Ric = 0 and (1.3), we can easily see thatrelation (1.5) holds true when relation (1.1) holds, and vice versa. From now on, a vector field A♯ on a quasirecurrent manifold (Mn, g) denotesa vector field associated with the 1-form A in (1.5), i.e., g(A♯, X) = A(X).Theorem 2.4. Let (Mn, g) be a quasirecurrent manifold. Then s/n is an eigen-value of the Ricci tensor corresponding to the vector field A♯.P r o o f. By virtue of the second Bianchi identity, we have(∇XR)(Y, Z, U, V ) + (∇ZR)(X,Y, U, V ) + (∇Y R)(Z,X,U, V ) = 0,which using (1.5) yieldsA(X)W2(Y, Z, U, V ) +A(Z)W2(X,Y, U, V ) +A(Y )W2(Z,X,U, V ) = 0.Contracting the last relation with respect to Y and V , from (2.1) we getA(X)( nn− 1)[Ric(Z,U)−sng(Z,U)]+A(Z)( nn− 1)[− Ric(X,U) +sng(X,U)]+W2(Z,X,U,A♯) = 0.Contracting the last relation with respect to Z and U , we have( 2nn− 1)[− Ric(X,A♯) +sng(X,A♯)]= 0,which yields(2.5) Ric(X,A♯) =sng(X,A♯).408Now we give some sufficient conditions for a quasirecurrent manifold to be Einstein.Theorem 2.5. Let (Mn, g) be a quasirecurrent manifold which is conformallyflat. Then the manifold is Einstein.P r o o f. It is well known [2] that a conformally flat manifold (Mn, g) satisfiesthe relation(∇XRic)(Y, Z)− (∇ZRic)(Y,X) =12(n− 1)[g(Y, Z) ds(X)− g(X,Y ) ds(Z)].From Theorem 2.1, it follows that the last relation yields(∇XRic)(Y, Z) = (∇ZRic)(Y,X).By virtue of (2.2), the last relation leads toA(X)( nn− 1)[Ric(Y, Z)−sng(Y, Z)]= A(Z)( nn− 1)[Ric(Y,X)−sng(Y,X)],which using X = A♯ and (2.5) yields‖A‖2nn− 1[Ric(Y, Z)−sng(Y, Z)]= 0.Hence, from the last relation and A 6= 0 it follows thatRic(Y, Z) =sng(Y, Z),showing that the manifold is Einstein. A Riemannian manifold is called a cyclic Ricci symmetric manifold provided thatits Ricci tensor Ric satisfies the relation:(2.6) (∇XRic)(Y, Z) + (∇ZRic)(X,Y ) + (∇Y Ric)(Z,X) = 0.Now we get the following theorem.Theorem 2.6. Let (Mn, g) be a quasirecurrent manifold which is of cyclic Riccisymmetry. Then the manifold is Einstein.409P r o o f. Taking account of (2.2) and (2.6), we have(2.7) A(X)[Ric(Y, Z)−sng(Y, Z)] +A(Z)[Ric(X,Y )−sng(X,Y )]+A(Y )[Ric(Z,X)−sng(Z,X)]= 0.In Walker’s lemma [11], it is said that if a(X) and b(X,Y ) are the numbers satisfyingb(X,Y ) = b(Y,X) and a(X)b(Y, Z)+a(Y )b(Z,X)+a(Z)b(X,Y ) = 0 for all X , Y , Z,then either all the a(X) are zero or all the b(X,Y ) are zero. Therefore from (2.7) andWalker’s lemma, we get either A(X) = 0 or Ric(Y, Z)− (s/n)g(Y, Z) = 0. However,A(X) = 0 is inadmissible by the defining condition of quasirecurrent manifold andhence we findRic(Y, Z)−sng(Y, Z) = 0,showing that the manifold is Einstein. A vector field V on a Riemannian manifold (Mn, g) is said to be torse-formingprovided that it satisfies(2.8) ∇XV = fX + ω(X)V,where f is a nonzero scalar and ω is a 1-form.Concerning a torse-forming vector field in a quasirecurrent manifold, we get thefollowing theorem.Theorem 2.7. Let (Mn, g) be a quasirecurrent manifold with nonvanishing scalarcurvature s. If its associated vector field A♯ is a unit torse-forming vector field, thenthe integral curve α of A♯ in Mn is geodesic.P r o o f. Taking account of (2.2) and (2.5), we have(2.9) (∇XRic)(Y,A♯) = A(X)nn− 1[Ric(Y,A♯)−sng(Y,A♯)]= 0.On the other hand, we get(2.10) (∇XRic)(Y,A♯) = ∇X [Ric(Y,A♯)]− Ric(∇XY,A♯)− Ric(Y,∇XA♯).From (2.5), (2.9), (2.10) and Theorem 2.1, it follows that0 =sng(Y,∇XA♯)− Ric(Y,∇XA♯),410which using torse-forming vector field A♯ yields(2.11) 0 =sn(∇XA)(Y )− f Ric(Y,X)− ω(X)Ric(Y,A♯).Putting Y = A♯ in (2.11), from (2.5) and g(A♯, A♯) = A(A♯) = 1 we get(2.12) 0 =sn(∇XA)(A♯)− fsnA(X)−snω(X).From (2.12), s 6= 0 and (∇XA)(A♯) = −A(∇XA♯), it follows that(2.13) A(∇XA♯) = −fA(X)− ω(X).However, from torse-forming vector field A♯ we know that∇XA♯ = fX + ω(X)A♯and hence(2.14) A(∇XA♯) = fA(X) + ω(X).Taking account of (2.13) and (2.14), we havefA(X) + ω(X) = 0.Putting X = A♯ in the last relation, we getf = −ω(A♯),which using torse-forming vector field A♯ yields∇XA♯ = −ω(A♯)X + ω(X)A♯.Putting X = A♯ in the last relation, we have∇A♯A♯ = 0,showing that the integral curve α of A♯ in Mn is geodesic. 4113. Quasirecurrent product manifoldsLet (Mn, g) be a Riemannian product manifold (Mp ×Mn−p, ĝ + g̃). In local co-ordinates, we adopt the Latin indices (or the Greek indices) for tensor componentswhich are constructed on (Mp, ĝ) (or (Mn−p, g̃)). Therefore, the Latin indices takethe values from 1, . . . , p, whereas the Greek indices run over the range p+ 1, . . . , n.A Riemannian product manifold (Mp ×Mn−p, ĝ + g̃) is called a quasirecurrent prod-uct manifold provided that the product manifold is quasirecurrent. Now we have thefollowing theorem.Theorem 3.1. Let (Mp ×Mn−p, ĝ + g̃) be a quasirecurrent product manifold.Then either one decomposition manifold (Mp, ĝ) is locally symmetric or the otherdecomposition manifold (Mn−p, g̃) is Einstein.P r o o f. Since any tensor components of R and its covariant derivatives withboth Latin and Greek indices together should be zero, from (1.5) we have0 = Rαβγδ;p = ApW2αβγδ,which leads to either(3.1) Ap = 0or(3.2) W2αβγδ = 0.Here semicolon “;” indicates covariant differentiation.In the case of Ap = 0, from (1.5) we haveRijkl;p = 0,showing that the manifold (Mp, ĝ) is locally symmetric.On the other hand, if we assume that Ap 6= 0, then from (1.3) and (3.2) we have(3.3) Rαβγδ +1n− 1(gαγ Ricβδ − gβγ Ricαδ) = 0.Contracting (3.3) over α, δ, we obtainnn− 1(Ricβγ −sngβγ)= 0,which impliesRicβγ =sngβγ ,showing that (Mn−p, g̃) is Einstein.Therefore, either one decomposition manifold (Mp, ĝ) is locally symmetric or theother decomposition manifold (Mn−p, g̃) is Einstein. 412Now we show a proper example of quasirecurrent manifold.E x am p l e 3.1. Let (Mn, gc) be a space of constant curvature. It is wellknown [2] that a space of constant curvature is Einstein. Therefore from (1.3) andRic = sg/n, we haveW2(X,Y, Z, U) = R(X,Y, Z, U) +sn(n− 1)(g(X,Z)g(Y, U)− g(Y, Z)g(X,U)),which leads to(3.4) W2 = 0because (Mn, gc) is a space of constant curvature.On the other hand, it is also well known [2] that a space of constant curvature islocally symmetric, that is,(3.5) ∇R = 0.Hence, both (3.4) and (3.5) imply that (1.5) holds for any nonzero 1-form A.Therefore (Mn, gc) is a quasirecurrent manifold.References[1] T.Adati, T.Miyazawa: On a Riemannian space with recurrent conformal curvature.Tensor, New Ser. 18 (1967), 348–354. zbl MR[2] A.L. Besse: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Springer, Berlin, 1987. zbl MR doi[3] E.Cartan: Sur une classe remarquable d’espaces de Riemann. I. Bull. Soc. Math. Fr. 54(1926), 214–264. (In French.) zbl MR doi[4] J.Kim: A type of weakly symmetric structure on a Riemannian manifold. Korean J.Math. 30 (2022), 61–66. zbl MR doi[5] G.P. Pokhariyal: Curvature tensors in a Lorentzian para Sasakian manifold. Quaest.Math. 19 (1996), 129–136. zbl MR doi[6] G.P. Pokhariyal, R. S.Mishra: Curvature tensors and their relativistics significance.Yokohama Math. J. 18 (1970), 105–108. zbl MR[7] G.P. Pokhariyal, S. K.Moindi, B.M.Nzimbi: W2-recurrent LP-Sasakian manifold. Uni-vers. J. Math. Math. Sci. 3 (2013), 119–128. zbl[8] H.S. Ruse: On simply harmonic spaces. J. Lond. Math. Soc. 21 (1946), 243–247. zbl MR doi[9] H.S. Ruse: On simply harmonic “kappa spaces” of four dimensions. Proc. Lond. Math.Soc., II. Ser. 50 (1948), 317–329. zbl MR doi[10] H.S. Ruse: Three-dimensional spaces of recurrent curvature. Proc. Lond. Math. Soc.,II. Ser. 50 (1948), 438–446. zbl MR doi[11] A.G.Walker: On Ruse’s spaces of recurrent curvature. Proc. Lond. Math. Soc., II. Ser.52 (1950), 36–64. zbl MR doiAuthor’s address: Jaeman Kim, Department of Mathematics Education, Kangwon Na-tional University, Chunchon 200-701, Kangwon Do, Republic of Korea, e-mail: jaeman64@kangwon.ac.kr.413

On quasirecurrent manifolds

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On quasirecurrent manifolds

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Institute of Mathematics AS CR, v. v. i.