Submersions of generic submanifolds of a Kaehler manifold

2 Submersions of Generic Submanifolds of a Kaehler Manifold Let .11 be all alIllost Hermitian manifold with almost complex structure .I and .11. a RieInannian manifold isometrically unnierseci in _ll. \Ve note that subnlanifolds of a Kaehler manifold are determined by the behaviour of tali , nt bundle of the sul>nlanifold under the action of t he almost a1111ple1 structure of the ambient. miianifokl. A si1hnhanifol(l ;1I is called holoIUorl)llic (coulplex) if J(1 ,(,11)) C 1,(:11) for revery p E Al. where 7(M) (leuotes the tangent space to .11 at the point p. .11 is called totally real if .I (7(.11)) C 71; (.11), for every p E .11, where 1r (.\I) denotes the normal space to .11 at the point p. As a generalization of holomnorpilic and totally real submamiifolds. CI?-suhnlanifolds were introduced by A. Be.lanco. A C11-sIm1nlallifold .11 of an almost Hermitian manifold ,1/ with an almost complex striIetilre J requires tWWWu, orthogonal couhlllenmelItry distributions D and DL defined on .11 such that D is invariant under .1 and Dl is totally real (A. Bejaneu. 1978: B. Y. ('hen. 1981a). There is yet another generalization of ('R-sulnnanifolds known as generic tiupl1ianifok1s (B. Y. Chen. 1981c). These snbnnahifolds are defined by relaxing the condition on the coinp1enlentary distribution of ho1olhorlhic distribution. Let .\I be a real subilianifold of all alms t Hermit ian manifold .1I and let DI, = TI,:'lt tl .JTI .lI he he iimaximal hololnorphic suhspace of '11,(:11). If D : p - V t , definesa sinooth twloulorpliie distribution on .11. then .11 is called a generic SilbI11aIllfOl(l of M. The complementary distribution D" of V is called purely real distribution on .11. A generic sobxuanifold is a CR-subJualifcl(1 if the purely real distribution on .11 is totally real. A purely real distribution D" on a generic subinanifohi Al is called proper if it is not totally real. A generic subnlanifoki is called proper if purely real c1istribittiou is proper. 19 2O Chapter 2. Submersions of Generic Suhmanifolds of it Naehler Manifold On the other hand, the study of the Riemannian submersion r : .11 -a 13 of a Riemannian manifold Al onto a Riemannian manifold B was initiated by B. O'Neill (1966). A submersion r naturally gives rise to two distributions on Al called the horizontal and vertical distributions respectively, of which the vertical distribution is always integrable giving rise to the fibers of the subiiicrsion which are closed sul.)nianifolds of Al. For it CR-subma.nifold 11.1 of it Kaehler manifold ,tR. t lie distribution Dl is integrable (A. Bejancu, 1976). S. Kobayashi (1987) observed the similarity between the total space of submersion 7r : Al -*B and the C/1-suhmanifold Al of a Kaehler manifold Al in terms of the distributions. Thus he considered submersion i : Al > B of a CR-submanifold Al of a Kaehler manifold Al onto an almost Hermitian manifold B such that the distributions D and D'of Al become respectively the horizontal and vertical distributions required by the submersion r and r restricted i o D become an isoniet r • which preserves the complex structures. that. is. Ja-r+ = ir.oJ ut, D where J and J' are t lie almost complex structures of Al and B respectively. He has shown that tuder this s;tuat ion 13 is necessat ilk• a Iiaehler manifold and obtained the relations between holomorphic sectional curvatures of Al restricted to D and that of B. Further this study has been extended by S. Deshmukh, S. Ali and S. 1. Husain (1988). in which they obtained the relations between the Ricci curvatures and the scalar curvatures of a Kaehler manifold and the base manifold. To deal with the similar question for the generic submanifold of a Kaehler rnanifold, one has the difficulty that the distribution V' for generic simbntauifold of it Kaehler manifold is not necessarily integrable to match the requirement of the submersion. To overcome this difficulty we consider the submersion r, :.11 —> B of generic submanifolds Al of a Kaehler manifold Al onto an almost Hermitian manifold B with the assumption that. D" is integrable. In the first. section of this chapter. we have considered submersion of a Cfl subinanifold of it Kaehler manifoldl onto an almost Hermitian manifold and ob- tain product, theorems on such submersion by imposing conditions on CR-submanifold and its distribution. In section 2 we study the submersions of generic submanifolds Al of a Kaehler manifold Al onto an almost Hermitian manifold B with integrable purely rail distribution D° and Drove that for the submersion of a generic submnanifold A! of a Kaeliler manifold Al onto an almost Hermitian manifold B. B is necessarily a Kaeliler manifold and obtain time decomposition theoremms for the generic submnanifold Al. Also we 2.1. Submersion of CR-Siihmanifolds 21 have obtained the relation between the holomorphic sectional curvatures of :1f1 restricted to D and that. of B. In the last of this chapter we have obtained the relation between the holornorphic sectional curvatures of :11 restricted to D and that of B with totally geodesic fibres. The contents of this chapter are partially published in Global Journal of Advanced Research on Classical and Modern Geometries. 2.1 Submersion of C Yf-Submanifolds In this section. we have studied the. CR-suhmersiutis of lKaehlcr manifolds and a tnunber of decomposition t.heoreins have been proved here. We start this section with the definition of the stibinersion of a CJ?-subiiianifold of an almost. Hcrmit.ian manifold. Definition 2.1.1. (S . Kobayashi. 1987). Let 1I be a C. 1l-submanifold of an almost Her- 'two ifuld .11 with distribution D and D- and the normal bundle T 1 11. By a .uhntersinn r :.1 -+ B onto an almost Hermitian nr.anifold B use mean. a Riemannian submersion 7r : M -~ B together •with the following conditions: (i) D is the kernel of ,r., that is ir,(D') = {0}, (ii) ,,.: D1, -> T (t,) B is a complex isometry for all 1, E .11. where T O B is the tangent space of B at T(p). (iii) J interchanges D1 and T'- .11 i.e.. JD= = T 1:11. ``fie call such submersion as CR-submersion. Let V. 0 and 0' denote Riemannian connections on R1, .11 and B respectively. For the c011i1e(lion C7" we define corresponding comiection V` for basic vector fields on Al by pt}'= 7-l(V'% Y). (2.1.1) Then V Y is a basic vector field. and by Lemma 1.4.1, we have r•('.r Y) = V *.y .Y.. (2.1.2) We define a tensor field C on Al by x Y = 7\Y - C(X, Y), (2.1.3) for any X, V E [("D). where C(X. Y) is the vertical part of V. Y, i.e., V(V. Y) _ It has 1een observed that. C is skew svintinet.ric and satisfies 22 Chapter 2. Submersions of Generic Siibrnanifolcls of it Kaehler Manifold C(X. I- ) 2 = V[X. Y]. (2.1.4) for any X. Y E F(D). Also for X E r(D) and 1' E 1,(D ), we define an operator A on ,ll by V.~•[%=V(V,vV)+A.t1. (2.1.5) where A. V is the horizontal part of V.t l.'. Since [V, X1 E F(D1) for any basic vector field X and V E I'(D1), we have = W(V. X) = A_,c1'. (2.1.6) 9(A 1', Y) = —g(V. C(X, S')). (2.1.7) The operator C and A are related iw X. Y E I,(D) and V E F(D - ). The operator C in (2.1.3) was introduced by S. Kobayashi (1987). For vertical vector fields an operator L was introduced by S. Deshniukh, S. Ali and S. I. Husain (1988) in the following manner: For U. i! E I'(D' ). we define L by V t;V = ntf l~'--L(U,V). (2.1.8) where ti - = V(Vt,V) and L(C'. V) For horizontal vector field X and vertical vector field t' we set \7 X = 7-I(Vt'X) +T-X. (2.1.9) where T'X = V(V v X). Moreover. if X is basic, [V. X) E 1'(D - ) for V E 1'(D1) and we obtain 7-L(V l' ) = R (V x V) = A,v V. IIence, for a basic vector field X and V E F(V1) \V(' have = AxV + T'X. The Operators (2.1.10) T and L are related by g(7 X. IV) = —g(L(V 11'), X). (2.1.11) Let. Rt be the curvature tensor corresponding to the connection C" of the base manifold B then R' and R are related by R(X. I', Z. H) = R'(X.. Y.. Z. H.) — 9(C(X. Z), C( 1', H)) + g(C(Y. Z). C(X, H) ± 2g(C(X. Y), C(Z, H)). (2.1.12) for the horizontal vector fields X. )'. Z and H on .11. Let .7r : Al —* B be a CR-submersion and X, Y E F(D). then from (1.1.3) we have 2.1. Siihmersion of CR-Snhmanifolds 23 which on using (1.2.1), (1.4.1) and (2.1.2) yields Ct.lY + lc(X. JY) + C(X..IY) = Jt I., + JC(X. Y) -- .111(X,1"). (2.1.13) Comparing vertical and normal parts. we have Lemma 2.1.1. Let X. V be horizontal vector field.,, in .11. Then (r) ('11 X../Y) = .Ilr(.l". Y). (ii) h(.l',.11') = .IC(X.V) Next. we have Lemma 2.1.2. Let .11 be a CR-subnianifold of a Kaehler manifold 111 and n be a CRSO)mc r.;rurt. 77ten = .JA.v V, for amj X. Y E F(D) an.d [' E F(D) Proof. Let X be a basic vector field. I E ri "P; and V E 1'(D- ). Then using (1.1.3), 1.2.1 .. _' IT rtnd I.~ tttu: 1.1.1 (ii). we have (i 41 V.1") = g(RV.i 1 .1)) =9([L\".t j +V~. .1 . I) = g(V 'J_l. Y) = g(JV, X. Y) _ -g(V l. JY) _ -g(1(X ). JY) _ -g(Ax l'..11) = g(JA1'.Y). which proves the result. From above lemma and (2.1.6). it follows that ~ ❑ 24 Chapter 2. Silhme.rsions of Generic Suhnianifolds of a Kaehler Manifold C(JX, Y) = —C(X. JY). (2.1.11) \Vllicll innirn prove the following identity C(.I X..IY) = C(X. Y). (2.1.15) Proposition 2.1.1. Let ill be a D-totally geodesic CR-sabntanifold of a Kaehler manifold M. Then the horizontal distribution of a C'R-submersion is integrable and totally geodesic. Proof. Proof follows from the Lemma 2.1.1, Lemma 2.1.2 and (2.1.6). ❑ Theorem 2.1.1. Let JI be a D-totally geodesic CR-submanifold of a Kadrler manifold 11. Let. r be a CR-submersion, of .11 onto an almost Hermitian manifold B. Then 111 is a locally product manifold Rll x _11,), where 411 is an ini'ariant subntanifold and Al2 is a totally real subm.anifold of Al. Proof. Since .11 is totally geodesic, then Leninia 2.1.1 yields C(X, 1') = 0 for X. Y E I'(D) and thus from the defining equation of C i.e., (2.1.3) it follows that V. Y E r(D), that. is, D is parallel. Further. since on a C1?-suhmanifold Al of a Kaehler manifold Al, Dl is integrable and totally geodesic then from a result of A. Bejancu [Theorem 4.4, p-9], D' is also parallel. Since both the distributions are parallel, then by De Hham's Theorem Al is the product manifold All x :112 , where .111 is the integral subnmamiifolds of D and 1112 is integral submanifold of D1. This completes the proof of the theorem. ❑ As a consequence of Theorem 2.1.1 and Theorem 1.3 of (S. Kohavashi, 1987) we have Corollary 2.1.1. Let Al be D-totally geodesic CR-subman.ifold of a Kaehler irtanifold Al. If ir : 111 —4 B be a CR-submersion. from Al onto an almost Hermitian manifold B. then H(X) = H'(X ), for X E where H and H" are respectively the holontorphic sectional eurvature.s of Al and B. In particular. if Al is a space of constant holomorphic sectional curvature C, then so is B. For the submersion of CR-totally geodesic submanifold. we prove the following Theorem 2.1.2. Let Al be a CR-totally geodesic foliate subianifold of a Kaehler manifold X11 and let it :.11 — B be a CR-submersion of Al onto an almost Hermitian manifold 2.1. Submersion of CR-Suimianifolds 25 B. Then .11 is a locally product 1111 X :111 , where M 1 is an invariant subnaanifold and :111 is totally real su.bmanifold of M. Proof From the proof of the Theorem 2.1.1, it follows t hat. D is parallel. Also M being C'R- tot ally geodesic, we have /t(.l". l') = 0, for a,iv X E F(D), i-' E 1(D'). (2.1.16) For any vector field V U' E F(D), from (1.1.3) we have —AJ1t-V + V JIV = JV,, IV + Jh(V, It). (2.1.17) Tu complete the proof of the theorem, we need the following result whose proof vi11 be denionstrated at the end of the proof of the theorem. Lemma 2.1.3. .4 C. R-suhmanifold .11 of a Kaehler manifold :lI is CR-totally geodesic if and only if ANI." E r(D~- ) for each V E r(D1 ) and N E F(T' AI). If we admit this lelnnia in (2.1.1 7) and put Ill' = N. we see that A.jt-t V E F(D-). and equating vertical part we have = .Ih(V, lt-'). which t hen yields Ct.Jlt' = Jv j-IV. (2.1.18) Heiicc. from (2.1.1 ) it 1ufluwti that V1•li' E F(D L.), i.e.. DI is parallel and this proves Li the theorem. Proof of the Lemma. Let .11 be it CR-totally geodesic, then by using Definition 1.3.8 and (1.2.3) we get 9(A.\,17. X) = 0. for each N E 1'(D), V E I'(D') and N E I'(T'. 1). This implies that A.N,V E l'(D' ). Conversely, suppose that An-V E 1,(D-1 ) for any V.' E F(D1) and N E F(Tl.l1). Let {N1 ' .2 ..., 2i _,,, } be a local ort.honormal frame of T11. when 2n = dim :11 and in = cline .11. Then for aiiv N E F(D). we have (I = q(A.v,,t'. X) — g(h(V, N). N,,). 1 < p < 2n — m. (2.1.19) Since h(V.N) E h(T i ..11), from (2.1.19) it follows that. h.(V, 1) = 0. Hence. \I is C.'!?totally geodesic. Next . we provc 26 Chapter 2. Submersions of Generic Subnianifolds of a Kaehler Manifold Proposition 2.1.2. Let Al be a CR-submanifold of a Kaehler manifold Al and rr be a CR-submersion of Al onto an almost Hermitian manifold 13. Then the fibres of 7r are totally geodesic submanifold of Al if and only if Al is CI?-totally geodesic. Proof. From (1.1.3), it. follows that —AJi U +V~ JV =.IL(U.U)+.//(u.U). for any U. V E F(D'). Using (1.4.1), and comparing horizontal and vertical parts, we get and 1(A.wU) _ —JL(U. V) (2.1.20) V(Ajl, U) _ —Jh(U. U). (2.1.21) From (2.1.20) it follows that the fibres are totally geodesic if ui(l only if A.Jr-U E F(D - ). which by Lemma 2.1.3 proves that the fibres are totally geodesic if and only if Al is C.II-totally geodesic. ❑ Theorem 2.1.3. Let Al be a D-totally geodesic CR-submanifold of a Kaehler manifold Al and rr : Al B be a CR-submersion of Al onto an almost Hermitian manifold B. Then Al is locally product manifold all x :112 if and only if fibres are. totally geodesic submanifold of Al, where All is an in.varian.t and .11;, is totally real submanifold of Al. Proof. From the proof of the Theorem 2.1.1, D is parallel. Further. for any U. V E F(V) and X E 1'(D) using (1.4.1) and (2.1.8) we have g(VtuV. X) = 9(L(U. V). X). From which it follows that D1 is parallel if and only if fibres are totally geodesic, which t hen proves the t heorelll. ❑ Theorem 2.1.4. Let. Al be a CR-subm.anifold of a Kaehler manifold :11 and 7r : Al —► B be a CR-submersion of Al onto an almost Hermitian manifold B. Then .11 is a locally twisted product manifold All x :112 if and only if (i) T-,X = —g(X, L(V, V))IIV II -2 V and (ii) C(a..JY) = 0, for any X. Y E F(D) and V E F(D1), where Al, is an invariant. submanifold and All is totally real submanifold of Al. 27 2.2. Submersions of Generic Submanifolds Proof. For an V. 11' E F(D') and X E T(D). from (1.1.3). (1.2.2). (2.1.11) we have g(VI - li.X) =9( — =1.nt l.JY) _ — g( (A.j« V'). JX) = g(JL(V. U/'). JX) = g(L(V. IV). X) This implies that Dl is totidly' 1lmI)111cal if and only it TI,X = —X(A)t'. (2.1.22) where f is sonic Em1et ion on .11. On taking inner product with [' in (2.1.22) it follows that Using (2.1.1 1), «-e have .V(,A) _ — g(T .A'. l i ll 2 X(\) = g(L(V 1 ')• -1)Ij VI1 -2 Putting it, in (2.1.22), we get 7X _ -q(L(V ['). ,t)IIV(f`['. The proof then follows from Lemma 2.1.1. ❑ 2.2 Sl1bYllel'sioris of Generic Submanifolds In this section we define the submersion of it generic suMnanifold of it Koehler manifold onto an almost. Hermitian mmmanifold and (discuss the impact of such 5ul)1IlerS1onS on the geoinetry of generic subbnanifold .11. Definition 2.2.1. Lt t .11 be a generic subrnanifold of an almost Hermitian manifold , l with distributions D and D° and the normal bundle T':11. We assume that (i) D° is the kernel of r,, that is, 7r.(D° ) = {0}. (ii) r. (V,,) = T7 .( ) 13 is a complex isonaetmi, where pE :11 and T ( ,,)13 is the tangent .space of B at r(p). ?how we have the following lemma: Lemma 2.2.1. Let r :.11 —+ B he a submersion of generic subnaanifold M of a Koehler manifold .i7 onto an almost Henn-tican manifold B. Then 28 Chapter 2. Submersions of Generic Sulmianifolds of a Kaehler Manifold C(X. IV) = PC(X. Y) + th(X. Y), h(X. JY) = FC(X. Y) + f h (X, Y), for the horizontal vector field. X. }'. Proof. Since 111 is a Kachler manifold, we have for all X. Ye 1'(D). By using (1.2.1). (1.:3.1), (2.1.1) and (2.1.3), we have V JY + C(X, .JY) + h(X, JY) _ .I (V Y + C(X, Y)) + Jh(X, Y) =JV }'+PC(X,Y)+FC(X.Y) + th(X, Y) + f h(X, Y) (2.2.1) By comparing the horizontal, vertical and normal parts. we get (2.2.2) C(X, .JY) = PC'(X, Y) -f th(X. Y) (2.2.3) h(X, .1Y) = FC(X, Y) + fli(X, Y) (2.2.4) Hence the result. 0 From above leninia, we have the following proposition: Proposition 2.2.1. Let r : M —> B be a submersion of generic submanifold Al of a Kachier manifold Al onto an. almost Hermitian -manifold B. Then C'(X, JY) + h(X, .JY) = JC(X. Y) + Jh(X. Y). for all X. V E Lemma 2.2.2. Let Al be a generic submanifold of a Kaehler manifold Al and r ::1i — B be a submersion from generic submaniiold Al onto an almost Hermitian manifold B. Then L(V, PH') — R(AF1 V) = JL(1', li") (2.2.5) V v-P[t' — V(A1.'jt•V) = PV U" + th(V, lt') (2.2.6) h(V, Plt') + V FI ,V = FV,-1F + f h(V, W), (2.2.7) 2.2. Submersions of Generic Subrnanifolds 29 for any vevli a& vector fields V. W. Proof. Since M is a Kachler manifold, we have V v Jl+ = JVW, for any V, W E F(D1)_ By using (1.3.1), we get c' PU_+V,FW =JDv W. By using Gauss and ½Veingar9. en Iormulae, we have VPWW'+h(V, PW) +(—AFwV)+ VFW = J(VW + h(V, W)). (2.2.8) Further on using (1.3.1), (1.3.2), and (2.1.8) in (2.2.8), we get. L(V, PW) +VyPW + h(V,PW) —'H(AFwV) —V(AF-wV) = ✓(V v W +VTFW + L(V. W) +h(v,W)) =PVvW+FVv IV+JL(V,W) + th(V W) + fh(V, W). (22.9) By comparing horizontal, vertical and normal parts in (2.2.9), we get L(V,PW) - N(Ar wV) _ JL(V,W) wPW — V(AV) = POvW + th(V, W) h(V, PW) — VFW — FwW+ fh(V, W). Hence the result. 70 Proposition 2.2.2. Let ht be a gen.erie submanifold of a Kaehter manifold M avd rr : Al —r B be a subncersion. franc generic subtuanifold Al onto an almost Henrcetian ncanifaid B. Then A ✓xV — J.AA v , for any X. Y E P(P) and V E I'(a° ) Proof, Similar to proof of Leninia 2.1.2. rrom Proposition 2.2.2 and (2.L@) i[ follows that. Proposition 2.2.3. Let Al be a generic atü,timifo1d of a Kaehler manifold M and s 69 - D be a submersion from gvua ,-ec submanifold M onlo an almost Hermitian manifold 30 Chapter 2. Submersions of Generic Subnianifolds of a Kaehler Manifold B. Then C(JX. JY) = C(X. }'), for all X, Y e F(D). As a consequence of the above result, we have the following corollary: Corollary 2.2.1. For horizontal vector fields X and Y. use have C(X. JY) = —C(.IX . Y). Proposition 2.2.4. .A generic submanifold If of a Kaehler manifold 11I is mixed totally geodesic if and only if ,fl.\•V E r(D° ) (respectively A,X E r(D)), for V E h(D° ) (respectively X E r(D)) and N E r(T1A1). Proof. Let .11 be a mixed totally geodesic, then by (1.2.3), we have g(A ', X) = g(h(X. V). N) = 0. for any X E r(D) and V E I'(D° ), which implies that A,~.V E r(D°) for V E r(D° ) Similarly, we can prove that A.%, X E r(D) for X E r(D). Conversely. suppose that A;vV E r(D° ) for any V E r(D° ) and N E r(T l lll ). Then for any X E F(D). q(A~'V,X) 0. Again h} using (1.2.3), we have g(h(X. Y). N) = 0, for N E r(T1:11). Since h(X, V) E r(T'Al), from (2.2.10) it follows that h(X. V) totally geodesic. Which completes the proof. (2.2.10) = 0, i.e., Al is mixed 0 Now. we prove Theorem 2.2.1. Let Al be a generic subm.anifold of a Kaehler manifold Al and it : Al — B he a submersion from generic submanifold .11 onto an almost Hermitian manifold B. Then. B is a Kaehler manifold. Proof. From (2.2.2) for any basic vector fields X and Y, we have Operating it on the above equation to project it down on B and using Leinnla 1.4.1, we get 2.2. Submersions of Generic Submanifolds V\..1'1: for any vector fields X,. Y. E F(TB), 31 — where J'V Y. r.X = X. and r,, Y' = Y. aahnost complex structnlrc on B. This proves that. B is it Kaehler manifold. and J' is the ❑ From the definition of Riemannian submersion r : Al —* B of a Riemannian manifold Al onto a Riemannian iWaIlifolcl 13, it follows that the vertical distribution is always integrable <<lul it.,, integral inanifoltt are thu fibers (B. ('Neill, 1966). which are closed Silhmunifold s of ,11. If in (il(iition, Do' is parallel, shell we have Theorem 2.2.2. Let al be a generic submanifold of a Kaehlcr manifold :11 and 7r ::11 —3 B btu a ,otbi ision from .11 onto an almost Hermitian manifold 13. If D is integrable and V° is parallel, then _lI is the Riernannian product. Ali x AI-~, where film is an in.vari.crnt subrnanifold and .Al2 is a purely real subnianifold of Al. Proof. Since the horizontal (list.ribut.ion D is integrable, so [X. Y] E F(D), for X. Y E ['(D): therefore V[X.1". = U. Then from (2.1.5). we have C'(X.Y) = 0. Titus from the definition of C. We have Cr_y' — n Y E F(D), that. is D is parallel. Now, since the horizontal listrihnt.ion D and vertical distribution D° are both parallel, it means that both the distributions D and D° are integrable and their leaves are totally geodesic in .11. therefore by Dc Rhamu's Theorem, it follows that Al is the generic pro(luct., i.e., .t/ _ tai x .112, where it1 ' and alp are integrable suhnlxnifold of D and D° respectively. From I lit, properties of D and D". it follows that .11k is invariant sublii nifold of .11 and ,lfi is I)iucelr real suhnlanifold of Q. ❑ Let N and N denote the holomorphic• sectional curvatures of :1/ and B respectively. I11 order to compare the holonlorphic sectional curvatures of :lt with that of B, we calculate the bisectional curvature. For this, we set Z = .I1 F. Y = JX in (1.2.4) and (2.1.12) and get l(t',.Jsh", X..I,I") = R(it,Jt''..IX)+g(h(X',.Ill'),h(JV.IF)) — gh(.JX..IIi) h(.1", IF)) (2.2.11) R(IF,.1i1". X, IX) = B'([l..I'lt',X,.J'X> ) —q(C(JX,.1nW'),C(X.It')) -- g(C(X, .JIF), C.(.Ll", IF)) + 2g(C(X, IX). C(JtV. GW/)), (2.2.12) 32 Chapter 2. Submersions of Generic Submanifolds of a Kaehler Manifold for any basic vector fields X. Y. Z. and W on 11. From (2.2.11) and (2.2.12), we have J(1 ; JU", X, IX) = R'(U , .I'U,X., J'X # ) + g(h(X..I11 "), h(.I.Y.14/ , )) — g(h(.IX, .J11'). h(X,1i1 )) — g(C(J_l..J11 ). C(X. [[')) + g(C(X, Jit'), C(.I,k', IV)) + 2g(C'(X, IX). C'(L[", J[i')). (2.2.13) From above equation, we have the following theorem: Theorem 2.2.3. Let al be a Kaehler manifold and Al be a generic' subrnanifold of Al with D integrable. Let B be an almost Hermitian manifold and it : Al B be a submersion then the bisectional curvatures K and l of X11 and B n'spectivehl satisfy K (IV, X) = K-(U -.. X.) + 1111.(X..JI1 , ) 11 2 + IIh(X, IV) 12. for any X. 1[" in r(D). Proof. Since D is integrable then h(X, J117 ) = h(JX.1U), (2.2.14) for any X. IF E r(D). Also. from Proposition 2.2.2 we have C'(X,I - )=0, (2.2.15) for any X. Y E h(D). Using the relation (2.2.14) and (2.2.15) iri (2.2.13), we get. the result.. In order to compare the holomorphic sectional curvatures of Al and B. we have the following theorem whose proof follows from Theorem 2.2.3. Theorem 2.2.4. Let 7r be a submersion from a generic subrnanifold Al of a Kaehler manifold :11 onto an. almost Hermitian manifold B. If the horizontal distribution D is integrable, then the holornorphic sectional curvatures of :11 and B satisfy 7-1(X) = f'(X,) + Ilh(X,,I X)II2 + Ilh(X, X )112 . ( 2.2.16) for any X E F(D). From above result, we have the following corollary: Corollary 2.2.2. Let r be a submersion from generic subinanifold Al of a Kaehler manifold 111 onto an almost Hermitian. manifold B. If the horizontal distribution D is integrable, 2.3. Submersions of Generic Suhmanifolds wit It Totally Geodesic Fibres 111#1) 33 111(- lu)i()n10171hir Sectional (rrrrHlr1rr '-1 (Ind 1-L' (If 11 Mid 13 i'(•spC(•fil•ety. are equal if and only if If i.5 D-totatli1 yeode.src. 2. 3 Submersions of Gener ic Submanifolds with Totally Geodesic Fibres III tins sect ion we discuss the sul)utersio>n of geiieric subeuonifotcl of it Kaehler manifold onto an almost lkrniitian manifold with totally geodesic fibres and we assume that. v = 0. Definition 2.3.1. The fibres we totally geodesic if L(U. V) = U for any vertical vector !ivlds U. Proposition 2.3.1. Lct r be a ,submersion. from a generic .subinanrfold .11 of a Kachlcr Fn(1l1)fold .l1 onto an almost Hermitian manifold B. If the fibers are totally geodesic subnzmtifolds of .11 tlrert Al is mixed totally geodesic. i.e.. 11(X, V) = 0. for any X E F(D) and V E r(v°). Plvof. Since fibers are totally "co(le sic . 1.A.. /:(l".IV) = () for at 2.2.5) we have V, It E F(D°), then by ~l(Artsl') =0. ~Vliic•li iinlili(s tlha+t Now, for any X C F(D) using (1.2.3) we get 0 = gl(.4rW V. X) = 9(h(V. X). /•'ll 1 ❑hewfor e. l>, the non-degeneracy of g we get the result. ❑ Remark 2.3.1. The conit'crsc of the above result is also true for the submersion of C'R111)m )lr fol(l of n Koehler manifold (S. Deshmvkh. S. Ali and S. 1. Husain, 1.988). F the ett(itluwr!)Iiistn 1' : ['Al —~ T.11, we have (V r P)F = VE;PF — P(C'1; F) (2.3.1) for any vector fields E and 1' tangent to .11. The endomorphism P is said to be parallel, if VP = 0 or (~'EP)F = 0 for any vector X I tulgent to Al. By using (2.1.S) for any V. 11' E r(D1). we have 311 Chapter 2. Subuwrsirrns of Generic Sabnianifolds of a Naehler Manifold ('t, P)Fl• = V14Pli' i L(%'. I'll ') — P(V1.lt') — I'L(1'. It'). (2.3.2) Now, if we suppose that the fibers are totally geodesic then above equation gets the form (nt l')it' = V. PIl" — PC t it: (2.3.3) Further, if P is parallel then (2.3.3) yields = Plt• (2.3.4) If we tiow consider that the fiheN arc totally geodesic and P is parallel. then (2.2.5) and (2.2.6) of Lemma 2.2.2 give 7-L( 4 , t% , I') — 0 and Ar~w•V = th(V, \V), i.e., Ai:~~ V = Jhh(V, \V). (2.3.5) Nt av we have t he following proposition: Proposition 2.3.2. Let it be a submersion from a generic submanifotd .11 of a Kaehler manifold :11 onto an almost Hermitian manifold B. If the fibers of :1f are totally geodesic and P is 1wnrallel. thcrl the sectional curnaturps of :V and fibers are rrlated by K(U A 1%) — K (U A V) - Y([Ar•a,AF•ti- jEr, ii'), for any U. l' E I'(D°) Proof We define 1? by R((.. 'I/)II — . V1 ] U' — Vut•:If Now, P(U, V)li' = V1, 7 vJI'F — V(U.vl6t' =Uc•V't—V V,;ll'— V,a.tfl'l' =V1.(L(V,G[% )+ V1,Ft') — C't(L(1',V)+ 7'1,.II') Taking inner product with vertical vector field F in above relation, we get R(U, V, It F) — R(U. V. Ii', F) — 9(L(V• U), L(U, F)) + g(L(U, II'). L(V, F)). From (1.2.4) and above relation. we have = V. It'. F) — g(L(V. U'), L(U, F)) + g(L(U. It'). L(V,F)) R(U. V. It, F) R(U. — g(h(L: II'). h(U. F)) + g(h(U. II'). h(V. F)). 2.3. Submersions of Generic Subiu nifolds with 'Totally Geodesic Fibres 35 The above relit ion gives R(U. V. U. l') — R(U, V. U. l') — g(L(U, V). L(U, lV)) + g(L(U, U). L(V. V)) — g(li(U. t'), h(U, t')) +!I(h(U. U). h(IV; V•)). Since fibers are totally geodesic and P is parallel, using (2.3.5) and the fact V .AFVVU ill t lie a>hove egitxt io1 i. we get. R(U.i'.C',V) = R(U.V.U , V) — g(Art;V , ArUV)+g( 4 uU,A,:t•V) Ii (U A V) — K(U A V) — g(A,.'uV, Apt•C) + g(Al'uU. A,•t-V) — f ((i A V) — g(AFuA!.' cU. V) + 9(ArvAruU, V) K(U A V) — g(• f.U.AF% - U — AF -t•AF-uU, V) — K(U A V) — .4(.ArJ, Art ]U, l%). U 1 ,Iii" P1,0%,C.1 the i'.ilt. Proposition 2.3.3. Let „ he a .submersion fir»n a generic .cub7n.anifold ill of a Koehler inain>J'olri .11 v:,!i, art uhnu.,l Hermitian manifold 13 with totallrj geodesic fibers. Then I?( 1. V. Y. n') ' g((Vt•(')i X. Y). tt') + g(Axl'• .41.1V) — g(h(X. v'), h(V. tt')), for any ,\, Y C I'(D) and V. It" ` 1(D0) Proof. By the defi>:iti(,n ( ,1 U. it follows that. VxV%•%' -1 V t•V,yY ='H(Vix.►)1 ) t 7t- .I' — Vx(l'i7v~Y+T~Y)+Vv(l-tVxY+C(X , }')) — V'.y (T,.}' i -- Vt (?-1V.a 2') -F V►•C'(.K. Y). Taking inner product with It' E I'(D'). We get g(R(t' .V)Y. U') _ g(Tx.vY It') — g(x('H~ - Y). ti') - g(V (T ). It ' ) + g(ct (' V.~ V). IV) + g(V1,C(X, Y), U') g(7 - v Y. tl') — y(Tc. )'. ll') — r!(HC7'Y, Vxll') — y(V,y( 7' }'), 11') —. lt') + g((Vy C)(X. Y), IV) + g(C(71G't .\. r"). IV) + a(C(X.1'LVy ,Y), IV). (2.3.6) = 36 Chapter 2. Submersions of Generic Submanifolds of a Kaehler Manifold From (2.1.11). we have q(T,Y, It') — —g(L(V Its'), Y) By taking covariant differentiation in above equation wit Ii respect to X. we obtain 9(T Y, V.-ht') _ —g(VxY. L(l'. It')) — g(V.x L(V, tt'), Y) 9(0.. (Ti }'). Il') —y(VxY. L(V. It')) — 9((V 1.)(V. It'). Y) — q(L(Q(vxti'),IV),Y) — y(L(V.Q(Vxt )). Y) _ — g(vxY. L(V. II')) — 9(Vx-L(V. II'). Y) +g(Tr...1-)'. 11 ') --g(Ti,Y.Q(V.vII')) 1(Vx (T , }')• IV) _ —9(vxL(t'. It'), Y) — g(vx)'. L(V. II')) + y(T',.1.Y, II'). From (2.3.6) and (2.3.7). we get. R(V. X. 1', it') = g(Tv, %,Y. II') — g(Tv,yY. II;') + y(P(V 1,Y). V..x II') +g((VxL)(V. IV). Y) + y(V xY, L(V, ll'!) — 9(7'c 1.Y, lV) — g(PV, )', v►•II') + 9((V%'C")(X, Y'), IV) + g(C(7 V1-X, Y), IV') + 9(C(X, 3LV % -Y). U') — g((VxL)(V. II';. Y) + 9((vvC)(X. Y). It') + q(P(v ti -'), V.~.IV) — q(Tv,..0Y. It") + g(C(HVj-X, Y), It') + g(C'(X. "HV j-Y). IV) Using (2.1.6). we get I?(V. X. I ". It') = q((v x L)(l . Ii'). Y I + q((v,,C)(X. Y). It') + g(Avl,',A.rIt') +q(Y, L(VV%, X, U')) + g(AytV, ni.X) — q(A.v It', OyY) = 9((v-x L) (V IV), Y) + g((vv(j)(X, Y), it') + y(Ay l-", A.IV) — q(Tff1 Y, VM-X) + 9(A,-It'.7-(VvX) — g(A.J-►l'. H(v, )') i = g((V7 xL)(V• II'), Y) +.q((v►'C)(X.Y). It') (2.3.7) 2.3. Submersions of Generic Subivanifolds with Totally Geodesic Fibres +9(Ai- t,A. 11') 37 q(TsY.'T X) I y(A, IV. A., V) - g(A.i ►l'. A, V). where we have used the definition of T and A and ?l(VvX) = 7-((V,rV) as [V, XJ E r(D°). Thus. l? V. N. . 1V) _ .9((V L)(l%, IV). 'i') - y((Vt C)(X. Y). ►l ) — y(Tv X.T„ Y). ,v by usi:ig 2 (2.3.8) iu (1 2. !), we get R(X. l~. V. U') = tr((VxL)il%. 1 t'),Y)+J((Vt - C)(X. 1"), W) +y(A.vl'..A? IV)-9(T, X.T„ 1') -g(1)(X.11 ),It(V•Y)) - 9(h(.Y.V),Ir(1 . 11")). Since the fibers are totally geodesic. then q(7 ' X. Tit}") _ — 9(X, L(l'. VV Y)) = 0 and by Proposition 2.3.1 Al is mixed totally geodesic i.e., h(X, V) — 0 for any X E I (D) and V E F(D). Hence we get the result. Cl A-; an immediate consequence of Proposition 2.3.3. we have Corollary 2.3.1. Lcl r be a .submersion from generic submanifold .11 of a Kaehler mani,'nld a1 onto urn uhnost Hrrrnitian manifold 13. If the fibers an totally geodesic submanifolds of al their I:(X A i = g(li(X. X), ii(V V)) — IIAxVII1, for unit vectors .0 E F(D) and V