We consider Ricci solitons with a semi-symmetric non-metric connection. We find some properties, when the potential vector field is torse-forming. Applications to submanifolds are also given

Doğru, Yusuf

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University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 40, No 2 (2025), 375–385https://doi.org/10.22190/FUMI240402028DOriginal Scientific PaperON RICCI SOLITONS AND SUBMANIFOLDS WITH ASEMI-SYMMETRIC NON-METRIC CONNECTIONYusuf DoğruAir Training Command, 35450 Konak-İzmir, TurkeyAbstract. We consider Ricci solitons with a semi-symmetric non-metric connection.We find some properties, when the potential vector field is torse-forming. Applicationsto submanifolds are also given.Keywords: Ricci soliton, semi-symmetric non-metric connection, torse-forming vectorfield, quasi-Einstein manifold, hyper-generalized quasi-Einstein manifold.1. IntroductionA semi-symmetric connection is a linear connection on a Riemannian manifold(M, g) whose torsion tensor T is of the formT (X1, X2) = ϕ(X2)X1 − ϕ(X1)X2,where ϕ is a 1-form defined by ϕ(X1) = g(X1, U), and U is a vector field on M [12].Let ∇ be the Levi-Civita connection of a Riemannian manifold (M, g). Thesemi-symmetric non-metric connection ∇̃ (briefly SSNMC) is defined by(1.1) ∇̃X1X2 = ∇X1X2 + ϕ(X2)X1,where X1, X2 are vector fields on M [1]. Let R̃ and R denote Riemannian curvaturetensor fields of ∇̃ and ∇, respectively. Then from (1.1), it is easy to see thatReceived April 02, 2024, accepted: May 17, 2024Communicated by Uday Chand DeCorresponding Author: Yusuf Doğru. E-mail addresses: yusufdogru1@yahoo.com2020 Mathematics Subject Classification. Primary 35Q51, 53B05, 53C25c⃝ 2025 by University of Nǐs, Serbia | Creative Commons License: CC BY-NC-ND375ORCID ID:   Yusuf Doğru https://orcid.org/0000-0002-1875-0243376 Y. Doğru(1.2) R̃(X1, X2)X3 = R(X1, X2)X3 − α(X2, X3)X1 + α(X1, X3)X2,where α is a tensor of type (0, 2) of M given by(1.3) α(X1, X2) = (∇X1ϕ)X2 − ϕ(X1)ϕ(X2).Let R̃ic and Ric state the Ricci tensor fields of the connections ∇̃ and ∇, re-spectively. Then from (1.2), it is easy to see that(1.4) R̃ic = Ric− (n− 1)α,(see [1]).Let (M, g) be a Riemannian manifold. R. S. Hamilton [14] presented the Ricciflow for the first time as∂∂tg(t) = −2Ric(g(t)).The Ricci flow is an evolution equation for Riemannian metrics. There is a corre-spondence between Ricci solitons and self-similar solutions of Ricci flow. A smoothvector field υ on a Riemannian manifold (M, g) is considered to define a Ricci soli-ton [13], if there exists a real constant λ such that(1.5)12£υg +Ric = λg,where £υ denotes the Lie derivative operator in the direction of the vector field υ,Ric denotes the Ricci tensor field of (M, g).We denote Ricci soliton by (υ, λ). It is obvious that Ricci solitons are naturalgeneralizations of Einstein metrics, any Einstein metric gives a trivial Ricci soli-ton. A Ricci soliton (υ, λ) on a (semi)-Riemannian manifold (M, g) is consideredto be shrinking, steady or expanding according to λ is positive, zero or negative,respectively [13].Quite a few geometers have lately studied the geometry of Ricci solitons. Referto, for instance, [8,17,18] and the references therein. Ricci solitons on submanifoldshave also become a quite popular study subject. For such studies refer to, forexample, [3, 5, 8] and the references therein.In the present study, we consider some properties of Ricci solitons on Rieman-nian manifolds equipped with an SSNMC when the potential vector field is torse-forming with respect to an SSNMC. As recent studies on torse-forming vectorfields see [?, 5, 9, 16].The paper is organized as follows: In Section 2, we discover the geometric prop-erties of Ricci solitons on Riemannian manifolds when the potential vector field istorse-forming. In Section 3, we get some applications for submanifolds.On Ricci Solitons and Submanifolds With a Semi-symmetric Non-metric Connection 3772. Ricci solitons on Riemannian manifolds with an SSNMCIn this section, we consider Ricci solitons on Riemannian manifolds with an SSNMC.The Euclidean 3-space, hyperbolic 3-space and Minkowski motion group areincluded in the following 3-parameter family of Riemannian homogeneous spaces(R3, g [µ1,µ2, µ3]) with left-invariant metricg [µ1,µ2, µ3] = e−2µ1tdx2 + e−2µ2tdy2 + µ23dt2.Here µ1,µ2 are real constants and µ3 is a positive constant.The Lie group G(µ1,µ2, µ3) can be realised as a closed subgroup of affine trans-formation group GL3Rn R3 of R3.The Levi-Civita connection ∇ of G(µ1,µ2, µ3) is given by the following formula:(2.1) ∇E1E1 =µ1µ3E3, ∇E1E2 = 0, ∇E1E3 = −µ1µ3E1∇E2E1 = 0, ∇E2E2 =µ2µ3E3, ∇E2E3 = −µ2µ3E2,∇E3E1 = ∇E3E2 = ∇E3E3 = 0.The Ricci tensor field Ric of G is given byR11 = −µ1(µ1 + µ2)µ23, R22 = −µ2(µ1 + µ2)µ23, R33 = −µ21 + µ22µ23and the scalar curvature τ of G is given byτ = − 2µ23(µ21 + µ22 + µ1µ2).(see [15]).Using (2.1), the Levi-Civita connection ∇ of G(1, 1, 1) is given by the followingformula:∇E1E1 = E3, ∇E1E2 = 0, ∇E1E3 = −E1,∇E2E1 = 0, ∇E2E2 = E3, ∇E2E3 = −E2,∇E3E1 = ∇E3E2 = ∇E3E3 = 0.Then we can state the following example:Example 2.1. Assume that υ = 2E3 is the potential vector field. Then G(1, 1, 1) is anRicci soliton with respect to an SSNMC.378 Y. DoğruUsing (1.1), we reach the Lie derivative as follows(2.2) (£̃υg)(X1, X2) = g(∇̃X1υ,X2) + g(X1, ∇̃X2υ)− 2ϕ(υ)g(X1, X2),Therefore, using equation (2.2), the soliton equation (1.5) with respect to an SSNMCcould be written as(2.3)12(g(∇̃X1υ,X2) + g(X1, ∇̃X2υ))+ R̃ic(X1, X2) = (λ+ ϕ(υ)) g(X1, X2).A vector field υ on a Riemannian manifold (M, g) is called torse-forming [20], if∇X1υ = cX1 + ω(X1)υ,where c is a smooth function, ω is a 1-form and ∇ is the Levi-Civita connection ofg.Specifically, if ω = 0, then υ is called a concircular vector field [11] and if c = 0,then υ is called a recurrent vector field [18].Presume that U is a parallel unit vector field with respect to the Levi-Civitaconnection ∇. Using (1.1), we reach∇̃X1U = X1.Consequently, we obtain the proposition below:Proposition 2.1. Let (M, g) be a Riemannian manifold endowed with an SSNMC.If U is a parallel unit vector field with respect to the Levi-Civita connection ∇ then,U is a torse-forming potential vector field (briefly TF − PV F ) with respect to anSSNMC of the form ∇̃X1U = X1.A non-flat Riemannian manifold (M, g) (n ≥ 3) is called a hyper-generalizedquasi-Einstein manifold [19], if its Ricci tensor field is not likewise zero and providesRic = a1g + a2A⊗A+ a3 (A⊗B +B ⊗A) + a4 (A⊗D +D ⊗A) ,where a1, a2, a3 and a4 are scalars and A, B and D are non-zero 1-forms. If a4 = 0,then M is called a generalized quasi-Einstein manifold in the sense of Chaki [6]. Ifa3 = a4 = 0, then M is called a quasi-Einstein manifold [7]. If a2 = a3 = a4 = 0,then (M, g) is an Einstein manifold [4]. The functions a1, a2, a3 and a4 are calledassociated functions.Now let (M, g) be a Riemannian manifold equipped with an SSNMC and υ aTF − PV F with respect to an SSNMC on M . Then ∇̃X1υ = cX1 + ω(X1)υ. Soby (2.3), we can write(2.4) Ric(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2)−12{g(X1, υ)ω(X2) + g(X2, υ)ω(X1)} .Hence we have:On Ricci Solitons and Submanifolds With a Semi-symmetric Non-metric Connection 379Corollary 2.1. Let (M, g) be a Riemannian manifold endowed with an SSNMCand υ a TF −PV F with respect to an SSNMC on M . Assume that a 1-form η isthe g-dual of υ. Then (M, g) is a Ricci soliton (υ, λ) if and only ifRic(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2)−12{η(X1)ω(X2) + η(X2)ω(X1)} .Thus, we can state the following theorems:Theorem 2.1. Let (M, g) be a Riemannian manifold endowed with an SSNMCand υ a TF −PV F with respect to an SSNMC on M . Assume that a 1-form η isthe g-dual of υ. Then (M, g) is a Ricci soliton (υ, λ) if and only if M is a hyper-generalized quasi-Einstein manifold with associated functions (λ− c+ ϕ(υ)) , 0, 0and − 12 .Theorem 2.2. Let (M, g) be a Riemannian manifold endowed with an SSNMCand υ a TF − PV F with respect to an SSNMC on M. Assume that a 1-form η isthe g-dual of υ and ω = η. Then (M, g) is a Ricci soliton (υ, λ) if and only if M isa quasi-Einstein manifold with associated functions (λ− c+ ϕ(υ)) ,−1.If υ is a concircular potential vector field (briefly C − PV F ) with respect to anSSNMC, then the following theorem can be expressed:Theorem 2.3. Let (M, g) be a Riemannian manifold endowed with an SSNMCand υ a C − PV F with respect to an SSNMC on M . Assume that a 1-form ηis the g-dual of υ. Then (M, g) is a Ricci soliton. (υ, λ) if and only if M is anEinstein manifold with an associated function (λ− c+ ϕ(υ)) .Using (1.4), the equation (2.4) can be written as(2.5) R̃ic(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2)− (n− 1)α(X1, X2)−12{g(X1, υ)ω(X2) + g(X2, υ)ω(X1)} .Thus, the following corollary can be expressed:Corollary 2.2. Let (M, g) be a Riemannian manifold endowed with an SSNMCand υ a TF − PV F with respect to an SSNMC on M . Then (M, g) is a Riccisoliton if and only if the Ricci tensor field of an SSNMC is of the form (2.5).380 Y. DoğruNow assume that U is a parallel unit vector field with respect to the Levi-Civitaconnection, i.e., ∇U = 0 and ∥U∥ = 1. Then(∇X1ϕ)X2 = ∇X1ϕ(X2)− ϕ(∇X1X2) = 0.So from (1.3), α(X1, X2) = −ϕ(X1)ϕ(X2). Thus by (2.5), we have(2.6) R̃ic(X1, X2) = (λ− c+ ϕ(υ))g(X1, X2) + (n− 1)ϕ(X1)ϕ(X2)−12{g(X1, υ)ω(X2) + g(X2, υ)ω(X1)} .Hence we have:Corollary 2.3. Let (M, g) be a Riemannian manifold endowed with an SSNMC,U a parallel unit vector field with respect to the Levi-Civita connection ∇ and υ aTF − PV F with respect to an SSNMC on M . Then (M, g) is a Ricci soliton ifand only if the Ricci tensor field of an SSNMC is of the form (2.6).Thus, the following theorems can be stated:Theorem 2.4. Let (M, g) be a Riemannian manifold endowed with an SSNMC,U a parallel unit vector field with respect to the Levi-Civita connection ∇ and υ aTF − PV F with respect to an SSNMC on M . Assume that a 1-form ϕ is theg-dual of υ. Then (M, g) is a Ricci soliton (υ, λ) if and only if M is a general-ized quasi-Einstein manifold with respect to an SSNMC with associated functions(λ− c+ ∥υ∥2), (n− 1) and − 12 .Theorem 2.5. Let (M, g) be a Riemannian manifold endowed with an SSNMC,U a parallel unit vector field with respect to the Levi-Civita connection ∇ and υ aC−PV F with respect to an SSNMC on M . Assume that a 1-form ϕ is the g-dualof υ. Then (M, g) is a Ricci soliton (υ, λ) if and only if M is a quasi-Einsteinmanifold with respect to an SSNMC with associated functions(λ− c+ ∥υ∥2)and(n− 1) .3. SubmanifoldsLet(M̃, g̃)be an (n + d)-dimensional Riemannian manifold endowed with anSSNMC ∇̃ and the Levi-Civita connection ∇. Decomposing the vector field U onM uniquely into its tangential and normal components UT and U⊥, respectively,we haveU = UT + U⊥.Let M be an n-dimensional submanifold of(M̃, g̃). On the submanifold M, letus denote the induced SSNMC by◦∇̃ and the induced Levi-Civita connection by◦∇.On Ricci Solitons and Submanifolds With a Semi-symmetric Non-metric Connection 381The Gauss formulas and Weingarten formulas with respect to ∇ and ∇̃ couldbe written as follows:∇X1X2 =◦∇X1X2 + h(X1, X2),∇̃X1X2 =◦∇̃X1X2 +◦h(X1, X2), X1, X2 ∈ χ(M),and∇X1N = −ANX1 +◦∇⊥X1N,∇̃X1N = −◦ANX1 +◦∇̃⊥X1N,respectively, where X1, X2 ∈ χ(M), h is the second fundamental form, N is a unitnormal vector field and AN is the shape operator of M in(M̃, g̃)and◦h is a normalvalued (0, 2)-tensor field and◦A is a (1, 1)-tensor field on M [2]. The tangential andnormal parts of U are denote by UT and U⊥, respectively. Then from [2], we get(3.1)◦h(X1, X2) = h(X1, X2)and(3.2)◦ANX1 = ANX1 − ϕ(N)X1.It is known from [2] that the induced connection◦∇̃ on a submanifold of a Rieman-nian manifold endowed with an SSNMC is also an SSNMC.Now assume that (M̃, g̃) is a Riemannian manifold admitting an SSNMC andυ is a TF −PV F with respect to an SSNMC on M̃ . Let (M, g) be a submanifoldof (M̃, g̃). Denote by υT and υ⊥, the tangential and normal parts of υ, respectively.Then using (1.1), it can be written as follows∇̃X1υ = ∇̃X1(υT + υ⊥)= ∇̃X1υT + ∇̃X1υ⊥= ∇X1υT + ϕ(υT )X1 +∇X1υ⊥ + ϕ(υ⊥)X1= cX1 + ω(X1)υT + ω(X1)υ⊥.So by using Gauss and Weingarten formulas and by the equality of the tangentialand normal parts, we have(3.3)◦∇X1υT = (c− ϕ(υ))X1 +Aυ⊥X1 + ω(X1)υTandh(X1, υT)+◦∇⊥X1υ⊥ = ω(X1)υ⊥.382 Y. DoğruThen in view of (3.3), we get(£υT g)(X1, X2) = g(◦∇X1υT , X2) + g(X1,◦∇X2υT )= 2 (c− ϕ(υ)) g(X1, X2) + 2g̃(h(X1, X2), υ⊥)+ω(X1)g(X2, υT ) + ω(X2)g(X1, υT ).Thus, equation (1.5) gives us(3.4)◦Ric(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2)− g̃(h(X1, X2), υ⊥)−12{ω(X1)g(X2, υT ) + ω(X2)g(X1, υT )}.So we reach the following corollary:Corollary 3.1. Let M be an n-dimensional submanifold isometrically immersedinto a Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ a TF−PV Fwith respect to an SSNMC on M̃ . Then (M, g) is a Ricci soliton (υT , λ) if andonly if the condition (3.4) holds on M .If M is υ⊥-umbilical, then Aυ⊥ = kI, where k is a function on M and I isthe identity map [8]. Therefore, we obtain g̃(h(X1, X2), υ⊥) = g(Aυ⊥X1, X2) =kg(X1, X2). Then from (3.4), we have(3.5)◦Ric(X1, X2) = (λ− c+ ϕ(υ)− k) g(X1, X2)−12{ω(X1)g(X2, υT ) + ω(X2)g(X1, υT )}.Hence we can state the following corollary:Corollary 3.2. Let M be an n-dimensional υ⊥-umbilical submanifold isometri-cally immersed into a Riemannian manifold(M̃, g̃)endowed with an SSNMCand υ a TF − PV F with respect to an SSNMC on M̃ . Then (M, g) is a Riccisoliton (υT , λ) if and only if the condition (3.5) holds on M .Thus, we can state the following theorems:Theorem 3.1. Let M be an n-dimensional υ⊥-umbilical submanifold isometricallyimmersed into a Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ aTF−PV F with respect to an SSNMC on M̃ . Assume that a 1-form η is the g dualof υT . Then (M, g) is a Ricci soliton (υT , λ) if and only if it is a hyper-generalizedquasi-Einstein manifold with related functions (λ− c+ ϕ(υ)− k) , 0, 0 and − 12 .On Ricci Solitons and Submanifolds With a Semi-symmetric Non-metric Connection 383Theorem 3.2. Let M be an n-dimensional υ⊥-umbilical submanifold isometricallyimmersed into a Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ aTF −PV F with respect to an SSNMC on M̃ . Accept that a 1-form ω is the g dualof υT . Then (M, g) is a Ricci soliton (υT , λ) if and only if it is a quasi-Einsteinmanifold with associated functions (λ− c+ ϕ(υ)− k) , −1.Since the induced connection◦∇̃ on a submanifold of a Riemannian manifoldendowed with an SSNMC is also an SSNMC, from (1.4), (3.1) and (3.4), we alsohave(3.6)◦R̃ic(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2)− (n− 1)α(X1, X2)−g̃(◦h(X1, X2), υ⊥)− 12{ω(X1)g(X2, υT ) + ω(X2)g(X1, υT )}where◦R̃ic denotes the Ricci tensor of the induced SSNMC.So we get the following corollary:Corollary 3.3. Let M be an n-dimensional submanifold isometrically immersedinto a Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ a TF−PV Fwith respect to an SSNMC on M̃ . Then (M, g) is a Ricci soliton (υT , λ) withrespect to the SSNMC if and only if the condition (3.6) holds on M .If U is a parallel unit vector field with respect to the Levi-Civita connection ∇,then we get(3.7)◦R̃ic(X1, X2) = (λ− c+ ϕ(υ)) g(X1, X2) + (n− 1)ϕ(X1)ϕ(X2)−g̃(◦h(X1, X2), υ⊥)− 12{ω(X1)g(X2, υT ) + ω(X2)g(X1, υT )}.If M is υ⊥-umbilical, then by (3.2), we have◦Aυ⊥X1 =(k − ϕ(υ⊥))X1,which brings us(k − ϕ(υ⊥))g (X1, X2) = g(◦Aυ⊥X1, X2)= g̃(◦h(X1, X2), υ⊥).Hence from (3.7), we have◦R̃ic(X1, X2) = (λ− c− k + ϕ(υ) + ϕ(υ⊥)) g(X1, X2) + (n− 1)ϕ(X1)ϕ(X2)−12{ω(X1)g(X2, υT ) + ω(X2)g(X1, υT )}.So we get the following theorem:384 Y. DoğruTheorem 3.3. Let M be a υ⊥-umbilical submanifold isometrically immersed intoa Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ a TF−PV F withrespect to an SSNMC on M̃ . Assume that a 1-form ϕ is the g dual of υT . Then(M, g) is a Ricci soliton (υT , λ) if and only if it is a hyper-generalized quasi-Einsteinmanifold with respect to an SSNMC with associate functions(λ − c − k + ∥υ∥2+∥∥υ⊥∥∥2 ), (n− 1) , 0 and − 12 .When υ is a C−PV F with respect to an SSNMC on M̃ , we have the followingtheorem:Theorem 3.4. Let M be a υ⊥-umbilical submanifold isometrically immersed intoa Riemannian manifold(M̃, g̃)endowed with an SSNMC and υ a C−PV F withrespect to an SSNMC on M̃ . Assume that a 1-form ϕ is the g dual of υT . Then(M, g) is a Ricci soliton (υT , λ) if and only if it is a quasi-Einstein manifold withrespect to an SSNMC with associate functions(λ− c− k + ∥υ∥2 +∥∥υ⊥∥∥2) and(n− 1) .REFERENCES1. N. S. Agashe and M. R. Chafle: A semi-symmetric non-metric connection on aRiemannian manifold. Indian J. Pure Appl. Math. 23(6)(1992), 399–409.2. N. S. Ageshe and M. R. Chafle: On submanifolds of a Riemannian manifold witha SSNMC. Tensor (N.S.) 55(2) (1994), 120–130.3. H. Al-Sodais, H. Alodan and S. 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ON RICCI SOLITONS AND SUBMANIFOLDS WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

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ON RICCI SOLITONS AND SUBMANIFOLDS WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

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