CUBO A Mathematical Journal Vol.21, No¯ 02, (41–49). August 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000200041 Totally umbilical proper slant submanifolds of para-Kenmotsu manifold M.S. Siddesha1, C.S. Bagewadi2 and D. Nirmala3 1Department of mathematics, New Horizon College of Engineering, Bangalore, India 2,3Department of Mathematics, Kuvempu University, Shankaraghatta, Shimoga, Karnataka, India mssiddesha@gmail.com, prof bagewadi@yahoo.co.in, nirmaladraj14@gmail.com ABSTRACT In this paper, we study slant submanifolds of a para-Kenmotsu manifold. We prove that totally umbilical slant submanifold of a para-Kenmotsu manifold is either invariant or anti-invariant or dimension of submanifold is 1 or the mean curvature vector H of the submanifold lies in the invariant normal subbundle. RESUMEN En este paper estudiamos subvariedades inclinadas en variedades para-Kenmotsu. De- mostramos que una subvariedad inclinada en una variedad para-Kenmotsu totalmente umbilical es invariante, o anti-invariante, o una subvariedad de dimensión 1, o el vector de curvatura media H de la subvariedad vive en el fibrado normal invariante. Keywords and Phrases: para-Kenmotsu manifold; totally umbilical; slant submanifold. 2010 AMS Mathematics Subject Classification: 53C25, 53C40, 53D15. http://dx.doi.org/10.4067/S0719-06462019000200041 42 M.S. Siddesha, C.S. Bagewadi and D. Nirmala CUBO 21, 2 (2019) 1 Introduction The study of submanifolds of an almost contact manifold is one of the utmost interesting topics in differential geometry. According to the behaviour of the tangent bundle of a submanifold with respect to action of the almost contact structure φ of the ambient manifold, there are two well known classes of submanifolds, namely, invariant and anti-invariant submanifolds. Chen [4], introduced the notion of slant submanifolds of the almost Hermitian manifolds. The contact version of slant submanifolds were given by Lotta [12]. Since then many research articles have been appeared on the existence of different contact and lorentzian manifolds (See. [1, 3, 7, 14, 15]). Motivated by the above studies, in the present paper we study slant submanifolds of almost para-Kenmotsu manifold and give a classification of results. Also we prove that totally umbilical slant submanifolds of para-Kenmotsu manifolds are totally geodesic. The paper is organized as follows: In section 2, we review some basic concepts of para-Kenmotsu manifold and submanifold theory. Section 3 is the main section of this paper. In this section we give the classification result of totally umbilical slant submanifolds of para-Kenmotsu manifold. Further, we prove that totally umbilical slant submanifolds of a para-Kenmotsu manifold are totally geodesic. 2 Preliminaries Let M̃ be a (2m + 1)-dimensional smooth manifold, φ a tensor field of type (1, 1), ξ a vector field and η a 1-form. We say that (φ, ξ, η) is an almost para contact structure on M̃ if [18] φξ = 0, η · φ = 0, rank(φ) = 2m, (2.1) φ2 = I − η ⊗ ξ, η(ξ) = 1. (2.2) If an almost paracontact manifold admits a pseudo Riemannian metric g of signature (m + 1, m) satisfying g(φ·, φ·) = −g + η ⊗ η (2.3) called almost para contact metric manifold. Examples of almost para contact metric structure are given in [6] and [9]. Analogous to the definition of Kenmotsu manifold [10], Welyczko [17] introduced para-Kenmotsu structure for three dimensional normal almost para contact metric structures. The similar notion called p-Kenmotsu structure appears in the Sinha and Sai Prasad [16]. Definition 2.1. An almost para contact metric manifold M(φ, ξ, η, g) is para-Kenmotsu manifold if the Levi-Civita connection ∇̃ of g satisfies (∇̃Xφ)Y = g(φX, Y)ξ − η(Y)φX, (2.4) CUBO 21, 2 (2019) Totally umbilical proper slant submanifolds . . . 43 for any X, Y ∈ χ(M), (where χ(M) is the set of all differential vector fields on M). From (2.4), we have ∇̃Xξ = X − η(X)ξ. (2.5) Assume M is a submanifold of a para-Kenmotsu manifold M̃. Let g and ∇ be the induced Riemannian metric and connections of M, respectively. Then the Gauss and Weingarten formulae are given respectively, by ∇̃XY = ∇XY + σ(X, Y), (2.6) ∇̃XN = −ANX + ∇ ⊥ XN, (2.7) for all X, Y on TM and N ∈ T⊥M, where ∇⊥ is the normal connection and A is the shape operator of M with respect to the unit normal vector N. The second fundamental form σ and the shape operator A are related by: g(σ(X, Y), N) = g(ANX, Y). (2.8) Now for any X ∈ Γ(TM) and V ∈ Γ(T⊥M), we write φX = PX + FX, (2.9) φV = pV + fV. (2.10) For X, Y ∈ Γ(TM), it is easy to observe from (2.1) and (2.9) that g(PX, Y) = −g(X, PY). (2.11) The covariant derivatives of the endomorphisms φ, P and F are defined respectively as (∇̃Xφ)Y = ∇̃XφY − φ∇̃XY, ∀X, Y ∈ Γ(TM̃), (2.12) (∇̃XP)Y = ∇XPY − P∇XY, ∀X, Y ∈ Γ(TM), (2.13) (∇̃XF)Y = ∇XFY − F∇XY, ∀X, Y ∈ Γ(TM). (2.14) The structure vector field ξ has been considered to be tangential to M throughout this paper, else M is simply anti-invariant [12]. Since ξ ∈ TM, for any X ∈ Γ(TM) by virtue of (2.5) and (2.6), we have ∇Xξ = X − η(X)ξ and σ(X, ξ) = 0. (2.15) Making use of (2.4), (2.6), (2.7), (2.9), (2.10) and (2.12)-(2.14), we obtain (∇̃XP)Y = pσ(X, Y) + AFYX + g(PX, Y)ξ − η(Y)PX, (2.16) (∇̃XF)Y = fσ(X, Y) − σ(X, PY) − η(Y)FX. (2.17) A submanifold M of an almost para contact metric manifold M̃ is said to be totally umbilical if σ(X, Y) = g(X, Y)H, (2.18) where H is the mean curvature vector of M. Further M is totally geodesic if σ(X, Y) = 0 and minimal if H = 0. 44 M.S. Siddesha, C.S. Bagewadi and D. Nirmala CUBO 21, 2 (2019) 3 Slant submanifolds of an almost contact metric manifold For any x ∈ M and X ∈ TxM such that X, ξ are linearly independent, the angle θ(x) ∈ [0, π 2 ] between φX and TxM is a constant θ, that is θ does not depend on the choice of X and x ∈ M. θ is called the slant angle of M in M̃. Invariant and anti-invariant submanifolds are slant submanifolds with slant angle θ equal to 0 and π 2 , respectively [5]. A slant submanifold which is neither invariant nor anti-invariant is called a proper slant submanifold. We mention the following results for later use. Theorem 3.1. [1] Let M be a submanifold of an almost contact metric manifold M̃ such that ξ ∈ TM. Then, M is slant if and only if there exists a constant λ ∈ [0, 1] such that P2 = −λ(I − η ⊗ ξ). (3.1) Further more, if θ is the slant angle of M, then λ = cos2θ. Corolary 1. [1] Let M be a slant submanifold of an almost contact metric manifold M̃ with slant angle θ. Then, for any X, Y ∈ TM, we have g(PX, PY) = −cos2θ(g(X, Y) − η(X)η(Y)), (3.2) g(FX, FY) = −sin2θ(g(X, Y) − η(X)η(Y)). (3.3) Theorem 3.2. Let M be a totally umbilical slant submanifold of a para-Kenmotsu manifold M̃. Then either one of the following statements is true: (i) M is invariant; (ii) M is anti-invariant; (iii) M is totally geodesic; (iv) dimM= 1; (v) If M is proper slant, then H ∈ Γ(µ); where H is the mean curvature vector of M. Proof. Suppose M is totally umbilical slant submanifold, then we have σ(PX, PX) = g(PX, PX)H = cos2θ{−‖X‖2 + η2(X)}H. By virtue of (2.6), one can get cos2θ{−‖X‖2 + η2(X)}H = ∇̃PXPX − ∇PXPX. From (2.9), we have cos2θ{−‖X‖2 + η2(X)}H = ∇̃PXφX − ∇̃PXFX − ∇PXPX. CUBO 21, 2 (2019) Totally umbilical proper slant submanifolds . . . 45 Applying (2.7) and (2.12), we get cos2θ{−‖X‖2 + η2(X)}H = (∇̃PXφ)X + φ∇̃PXX + AFXPX − ∇ ⊥ PXFX − ∇PXPX. Using (2.4) and (2.6), we obtain cos2θ{−‖X‖2 + η2(X)}H = g(φPX, X)ξ − η(X)φPX + φ(∇PXX + σ(X, PX)) +AFXPX − ∇ ⊥ PXFX − ∇PXPX. From (2.9), (2.11), (2.18) and the fact that X and PX are orthogonal vector fields on M, we arrive at cos2θ{−‖X‖2 + η2(X)}H = −g(PX, PX)ξ − η(X)P2X − η(X)FPX + P∇PXX + F∇PXX +AFXPX − ∇ ⊥ PXFX − ∇PXPX. Then applying (3.1) and (3.2), we obtain cos2θ{−‖X‖2 + η2(X)}H = cos2θ{‖X‖2 − η2(X)}ξ + cos2θη(X){X − η(X)}ξ − η(X)FPX +P∇PXX + F∇PXX + AFXPX − ∇ ⊥ PXFX − ∇PXPX. (3.4) Taking inner product with PX in (3.4), we get 0 = g(P∇PXX, PX) + g(AFXPX, PX) − g(∇PXPX, PX). (3.5) By virtue of (3.2), the first term of (3.5) can be written as g(P∇PXX, PX) = −cos 2θ{g(∇PXX, X) − η(X)g(∇PXX, ξ)}. (3.6) We simplify the third term of (3.5) as follows g(∇PXPX, PX) = g(∇̃PXPX, PX) = 1 2 PXg(PX, PX). = 1 2 PX[−cos2θ{(g(X, X) − η2(X))}] = − 1 2 cos2θ[PXg(X, X) − P(X)(g(X, ξ)g(X, ξ))] = − 1 2 cos2θ[PXg(X, X) − 2η(X)P(X)g(X, ξ)] = − 1 2 cos2θ[2g(∇̃PXX, X) − 2η(X){g(∇̃PXX, ξ) + g(X, ∇̃PXξ)}]. Using (2.5), (2.6), (3.6) and the fact that X and PX are orthogonal vector fields on M, we derive g(∇PXPX, PX) = −cos 2θ[g(∇PXX, X) − η(X)g(∇PXX, ξ) −η(X)g(X, PX − η(PX)ξ)] = −cos2θ[g(∇PXX, X) − η(X)g(∇PXX, ξ)] → g(∇PXPX, PX) = g(P∇PXX, PX). 46 M.S. Siddesha, C.S. Bagewadi and D. Nirmala CUBO 21, 2 (2019) Using this fact in (3.5), we obtain 0 = g(AFXPX, PX) = g(σ(PX, PX), FX). As M is totally umbilical slant, then from (2.18) and (3.2), we obtain 0 = −cos2θ{‖X‖2 − η2(X)}g(H, FX). (3.7) Thus from (3.7), we conclude that either θ = π 2 that is M is anti-invariant which part (ii) or the vector field X is parallel to the structure vector field ξ, i.e., M is 1-dimensional submanifold which is fourth part of the theorem or H ⊥ FX, for all X ∈ Γ(TM), i.e., H ∈ Γ(µ) which is the last part of the theorem or H = 0, i.e., M is totally geodesic which is (iii) or FX = 0, i.e., M is invariant which is part (i). This completes the proof of the theorem. Theorem 3.3. Every totally umbilical proper slant submanifold of a para-Kenmotsu manifold is totally geodesic. Proof. Let M be a totally umbilical proper slant submanifold of a para-Kenmotsu manifold M̃, then for any X, Y ∈ Γ(TM), we have ∇̃XφY − φ∇̃XY = g(φX, Y)ξ − η(Y)φX. Using equations (2.6) and (2.9), we get ∇̃XPY + ∇̃XFY − φ(∇XY + σ(X, Y)) = g(PX, Y)ξ − η(Y)PX − η(Y)FX. Again from (2.6), (2.7) and (2.9), we obtain g(PX, Y)ξ − η(Y)PX − η(Y)FX = ∇XPY + σ(X, PY) − AFYX +∇⊥XFY − P∇XY − F∇XY − φσ(X, Y). As M is totally umbilical, then g(PX, Y)ξ − η(Y)PX − η(Y)FX = ∇XPY + g(X, PY)H − AFYX + ∇ ⊥ XFY −P∇XY − F∇XY − g(X, Y)φH. (3.8) Taking the inner product with φH in (3.8) and from the fact that H ∈ Γ(µ), we obtain g(∇⊥XFY, φH) = −g(X, Y)‖H‖ 2. Applying (2.7) and the property of Riemannian connection the above equation takes the form g(FY, ∇⊥X φH) = g(X, Y)‖H‖ 2. (3.9) CUBO 21, 2 (2019) Totally umbilical proper slant submanifolds . . . 47 Now for any X ∈ Γ(TM), we have ∇̃XφH = (∇̃Xφ)H + φ∇̃XH. Using the fact H ∈ Γ(µ) and by virtue of (2.4), (2.7) and (2.9), we obtain − AφHX + ∇ ⊥ XφH = −PAHX − FAHX + φ∇ ⊥ XH. (3.10) Also for any X ∈ Γ(TM), we have g(∇⊥XH, FX) = g(∇̃XH, FX) = −g(H, ∇̃XFX). Using (2.9), we obtain g(∇⊥XH, FX) = −g(H, ∇̃XφX) + g(H, ∇̃XPX). Further from (2.6) and (2.12), we derive g(∇⊥XH, FX) = −g(H, (∇̃Xφ)X) − g(H, φ∇̃XX) + g(H, σ(X, PX)). Using (2.4) and (2.18), the first and last term of right hand side of the above equation are identically zero and hence by (2.3), the second term gives g(∇⊥XH, FX) = g(φH, ∇̃XX). Again by using (2.6) and (2.18), we obtain g(∇⊥XH, FX) = g(φH, H)‖X‖ 2 = 0. This means that ∇⊥XH ∈ Γ(µ). (3.11) Now, taking the inner product in (3.10) with FY for any Y ∈ Γ(TM), we get g(∇⊥XφH, FY) = −g(FAHX, FY) + g(φ∇ ⊥ XH, FY). Using (3.11) and from (3.3) and (3.9), we obtain g(X, Y)‖H‖2 = sin2θ{g(AHX, Y) − η(Y)g(AHX, ξ)}. (3.12) Hence by (2.8) and (2.18), the above equation reduces to g(X, Y)‖H‖2 = sin2θ{g(X, Y)‖H‖2 − η(Y)g(σ(X, Y), H)}. (3.13) Since for a para-Kenmotsu manifold M̃, σ(X, ξ) = 0 for any X tangent to M̃, thus we obtain g(X, Y)‖H‖2 = sin2θ{g(X, Y)‖H‖2. Therefore, the above equation can be written as cos2θg(X, Y)‖H‖2 = 0. (3.14) Since M is proper slant submanifold, thus from (3.14) we conclude that H = 0, i.e., M is totally geodesic in M̃. This completes the proof. 48 M.S. Siddesha, C.S. 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Bagewadi, Semi-slant submanifolds of (k, µ)-contact manifold, Com- mun. Fac. Sci. Univ. Ser. A1 Math. Stat., 67(2) (2017), 116-125. CUBO 21, 2 (2019) Totally umbilical proper slant submanifolds . . . 49 [16] B.B. Sinha and K.L. Sai Prasad, A class of almost para contact metric manifolds, Bull. Cal. Math. Soc., 87 (1995), 307-312. [17] J. Welyczko, Slant curves in 3-dimensional normal almost paracontact metric manifolds, Mediterr. J. Math., DOI 10.1007/s00009-013-0361-2, 2013. [18] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36(1) (2008), 37-60. Introduction Preliminaries Slant submanifolds of an almost contact metric manifold