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551 Internat. J. Math. & Math. Sci. Vol. I0, No. 3 (1987) 551-556 TOTALLY UMBILICAL CR-SUBMANIFOLDS OF SEMI-RIEMANNIAN KAEHLER MANIFOLDS K.L. DUGGAL and R. SHARMA Department of Mathematics and Statistics University of Windsor, Windsor Ontario, Canada N9B 3P4 (Received August 18, 1986 and in revised form October 20, 1986) ABSTRACT. We study totally umbilical CR-submanifolds of a Kaehler manifold carrying a semi-Riemannlan metric. It is shown that for dimension of the totally real distribution greater than one, these submanifolds are locally decomposable into a complex and a totally real submanifold of the Kaehler manifold. For dimension equal to one, we show, in particular, that they are endowed with a normal contact metric structure if and only if the second fundamental form is parallel. KEY WORDS AND PHRASES. CR-submanifolds, Totally umbilical, Semi-Riemannian metric, Kaehler manifold, normal contact metric structure, second fundamental form. 1980 AMS SUBJECT CLASSIFICATION CODE. I. 53C40. INTRODUCTION. The notion of a Cauchy-Riemann (CR)-submanlfold of a Kaehler manifold was introduced by Bejancu [1,2] and developed later by several researchers. detailed treatment, we refer to Yano and Kon [3]. far been confined to a positive definite metric. For a The study on CR-submanifolds has so The objective of this paper is to study totally umbilical CR-submanifolds of a Kaehler manifold carrying a semiRiemannian metric. According to Flaherty’s theorem [4], the Hermitian metric on a Kaehler manifold cannot have the Lorentzlan signature (which is essential for a spacetime manifold of relativity). This motivates us to consider CR-submanlfolds (whose metric can have the Lorentzlan signature) of a Kaehler manifold. 2. PRELIMINARIES. Let M be a submanifold isometrically immersed in a Kaehler manifold a complex structure We denote by the same Riemannian. J, a Hermitlan metric g M the metric of We also assume that TM the Levi-Civita connection induced in VxY VxY + B(X,Y) VxV -A + DxV g and M, which is assumed to be semi- is non-degenerate with respect to M M with and the Levi-Civlta connection g. If V is then we have (Gauss Formula) (2.1) (Weingarten Formula) (2.2) 552 K.L. DUGGAL AND R. SHARMA X,Y tangential to M and for arbitrary vector fields the second fundamental form, g(A,Y) g(X,Y), B(X,Y) particular, when B umbilical if g(B(X,Y),V). V D and M is the is said to be totally M In M. being the mean curvature vector field of 0, we say that B M; where normal to the shape operator associated to In fact, M. normal connection of V is totally geodesic. [I] of the Kaehler manifold M if there exists a dlfferentlable distribution D on M such that (I) D is invarlant by J, i.e. JD D for each x in M and (II) the complementary orthogonal distribution D x x is antl-varlant (totally real), i.e. JD for each x in is a subspace of T (M) x x M is said to be a CR-submanlfold A CR-manlfold is called proper if M. PX + FX, JX p3 + p FP 0, f3 O, + f 0, fF into their tangential and normal components (2.4a) 0 Pt O, tf (2.4b) 0 AX + yB(X,Y) (VxP)Y =-B(X,PY) + fB(X,Y) (VxF)Y (2.6) (Vxt)V AfvX PX (2.7) (Vxf)V -FA (2.8) B(X,tV) (VxP)Y Vx(PY) PVxY, (VxF)Y Dx(FY) FVxY (Vxt)V Vx(tV tDxV (Vxf)V Dx(fV fDxV PROPOSITION 2.1. Let PROOF. D M Let D the distributions D JV and (2.3) Following relations hold [3]: respectively. where JX and be a CR-submanifold of a Kaehler manifold. D1 0 for all Y Then there exists a non-zero vector field As D. in D X 0 because TM X This proves non-degenerateness of Y in is non-zero. D. X in are complementary and 0 for all But is non-degenerate. Dx and orthogonal to each other, it follows that g(X,Y) a contradiction. Then both are non-degenerate. be degenerate. such that g(X,Y) that Let tV + fV JV be the decompositions of # 0 and dlmDl # O. dlmD That Dx TM. This shows Hence we arrive at is non-degenerate can be proved likewise. PROPOSITION 2.2. The mean curvature vector to PROOF. JD1 in Consider any TMi. f in D g(J(VxX),JV) g(VxJX,JV) g(J(VxX),JV) g(VxX,V) 0. and of a totally umbilical CR-sub- V in the complementary orthogonal subbundle Then we have Thus we observe that i.e. X U JD1. manifold of a Kaehler manifold belongs to Hence g(X,X)g(,V) g(VxJX + g(VxX + 0. belongs to JI>L. g(X,JX)p,JV) g(X,X),V) O. g(X,X)g(,V). By prop. 2.1 it follows that g(,V) 0; TOTALLY UMBILICAL CR-SUBMANIFOLDS AFX LEMMA 2.1. (Yano-Kon [3]) AFyX, X,Y Y for any in Di. Let us The above lemma holds for both positive definite and indefinite metrics. dim D denote by q. q > I. TOTALLY UMBILICAL CR-SUBMANIFOLDS WITH 3. Then be a CR-manlfold of a Kaehler manifold. M Let 553 For a positive definite metric, the following is known: (Bejancu [5]). THEOREM 3.1. M. of a Kaehler manifold Let M be a totally umbilical proper CR-submanifold q > I, then M is totally geodesic in M and is If D locally a product of the leaves of Di, i.e. and a CR-product. That the above theorem holds also for an indefinite metric, can be proved as follows. XY AFyX for all X,Y xt tX. Since M By lemma 2.1, we have for any X in I>i we have PROOF. Di, in g(X,Y) B(S,Y) we have X in . g(,V)X. D Since q > Hence we obtain Di, that g(t,t) 0. f "A according to Chen’s theorem [6], M REMARK I. {i,2,3,4 M 0 on M for with 0. J 0 and hence Obviously we have CR-submanifold is a CR-product iff dim D dim D If we take Thus we get 0. VP 0. VP 0" This completes the proof. is a CR-product. p3 +p f-structure 0. Consequently therefore reduces to a totally geodesic submanifold. it follows that X t I, it follows, on contraction of (3.1) at Now, by prop. 2.2 we also have Thus lles t (3.1) respect to an orthonormal base of M As is totally umbilical g(t,t)X g(t,X)t for all X and Di. in 2 so that dim M 4 and assume that the is globally framed, then there exists a local basis PI 2’ P2 -I’ P3 P4 such that + + + 0. One can verify that g is Lorentzian with signature 3.1, M serves as a model of a class of decomposable space-times in general relativ- ity. For a recent study on such space-times we refer to [7]. TOTALLY UMBILICAL CR-SUBMANIFOLDS WITH q 4. Subject to the hypothesis of theorem I. Chen proved the following theorem: (Chen [8]). THEOREM 4.1. M. Kaehler manifold (i) M (ii) q (iii) M Let M be a totally umbilical CR-submanifold of a Then is totally geodesic, or I, or is totally real. Note that if M were a proper CR-submanifold in the above theorem, then the possibil- ity (iii) would be ruled out. Also note that (i) and (ii) are not mutually exclusive. The case (ii) has been investigated by Chen [8], in the context of a locally Hermitian symmetric space M and dim M 5. In the following theorem we study the case by relaxing these conditions and assuming THEOREM 4.2. manifold M. (I) M with Let q M to be proper. M be a proper totally umbilical CR-submanlfold of a Kaehler I. Suppose the mean curvature vector is non-vanishing over Then the following statements are equivalent: M has a normal contact metric (Sasakian) structure [3]. (2) has a non-zero constant norm. (3) is parallel in the normal bundle. K.L. DUGGAL AND R. SHARMA 554 (4) PROOF. and lles in t. M second fundamental form of For any # 0 and As Di. Now since X tangent to g(,)p2x P Operating is parsllel. JD lies in I, q t # 0 by prop. 2.2, it follows that any vector field in D is a scalar multiple of M we can show, using (2.7), that g(t,t)X g(t,X)t on this gives (4.1) g(t,t) g(,t) Hence we get g(t,t)(p2x + X) (4.2) g(t,X)t In this case too, equation (3.1) holds, which shows (q hence g(,) # 0. Equation (4.2) becomes p2X Di assume g(t,t) -a-2t a n and p2X 2 is a real-valued function on a , M. Under the setting equatln (4.3) takes the form + (X) P It follows that (4.3) is non-degenerate we can, without any loss of generality, where as the dual of -X g(t,t) # 0 and [g(t,t)]-Ig(t,X)t -X + As the distribution I) that (4.4) 0, rank (P) 0, hOP g(PX,PY) dimM- and (4.5) (X)(Y) g(X,Y) With the help of (2.5), (2.7) and (4.3) we derive (VxP)Y VX {g(,Y)X- g(X,Y)} a (4.6) (4.7) a PX Equations (4.4)-(4.7) show that M has a normal contact metric structure iff. 2 g(,) is a non-zero constant. This proves the equivalence of (I) to (2). By a virtue of the equality tD 2 X X in a t tDx the statement (2) is equivalent to 0. (obtained earlier), using (2.8) and operating f(Dx) DX Thus (2) is equivalent to 0. By differenting the result f2 f 0 on the derived equation we get 0, i.e. the statement (3). The state- ment (4) means Dx(B(Y,Z)) B(X,Y) Substitution + B(VxY,Z g(X,Y) B(Y,VxZ). in the above shows that (4) is equivalent to (3). This completes the proof. Let us compare theorem 4.2 with the following theorem of Chen [8]: REMARK 2. "Let M be M. If dim M (a) (b) a totally umbilical CR-submanifold of a locally Hermitlan symmetric space 5 and q I, then is parallel if M is not totally geodesic, I/a and rank M dim M + I. M Moreover, M is locally isometric to a sphere of radius is a totally umbilical hypersurface of a TOTALLY UMBILICAL CR-SUBMANIFOLDS flat, totally geodesic submanlfold of M." We observe that theorem 4.2 is consistent with the above mentioned theorem. the conclusion (b) of the theorem shows that (I) of theorem 4.2. ACKNOWLEDGEMENT. 555 M In fact, is Sasaklan, which is the statement Also the conclusion (a) is the statement (3) of theorem 4.2. We thank the Natural Sciences and Engineering Research Council of Canada for supporting with a research grant. We are also thankful to the referee for valuable suggestions for the improvement in the original version of this paper. REFERENCES I. BEJANCU, A. CR submanlfolds of a Kaehler Manifold I, Proc. Amer. Math. Soc. 69 (1978), 135-142. 2. BEJANCU, A. CR submanlfolds of a Kaehler Manifold II, Trans. Amer. Math. Soc. 250 (1979), 333-345. 3. YANO, K. and KON, M. CR submanlfolds of Kaehlerian and Sasaklan Manifolds, Birkhauser, Boston, 1983. 4. FLAHERTY, E.J. 5. BEJANCU, A. Umbilical CR submanlfolds of a Kaehler Manifold, Rendlcontl dl Mat. (3) 15 Serle VI (1980), 431-446. 6. CHEN, B.Y. CR-submanlfolds of a Kaehler Manifold. I, J. Diff. Geom. 16 (1981), 305-322. 7. DUGGAL, K.L. and SHARMA, R. Lorentzlan Framed Structures in General Relativity, Gen. Rel. Gray. 18 (1986), 71-77. 8. CHEN, B.Y. Totally Umbilical Submanlfolds of Kaehler Manifolds, Arch. der Math. 36 (1981), 83-91. Hermltlan and Kaehlerlan Geometry in Relativity, Lecture Notes in Physics, Springer Verlan, Berlin, 1976.