THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION

Transcription

1 Internatonal Electronc Journal of Geometry Volume 7 No. 1 pp (2014) c IEJG THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION AUREL BEJANCU AND HANI REDA FARRAN Dedcated to memory of Proffessor Frank Dllen Abstract. Let F m = (M, F ) be a Fnsler submanfold of a Fnsler manfold F m+p = ( M, F ). By usng the normal curvature vector feld of F m and the Berwald connectons on both F m and F m+p, we obtan the structure equatons for the mmerson of F m nto F m+p. Ths enables us to relate, for the frst tme n lterature, the flag curvatures of F m and F m+p. Fnally, we nvestgate the exstence of totally geodesc Fnsler submanfolds of a Randers (c, K)-sphere. Introducton The theory of Fnsler submanfolds s as old as Fnsler geometry s tself. Indeed, the Fnsler geometry has emerged n 1918 when Fnsler wrote hs thess on curves and surfaces n what he called generalzed metrc spaces. However, so far there s no well establshed theory of Fnsler submanfolds as we have the theory of Remannan submanfolds. Many reasons have concurred to ths state of affars n Fnsler geometry. Frst, we menton that the geometry of Fnsler manfolds s based on four classcal Fnsler connectons: Berwald connecton, Cartan connecton, Chern Rund connecton and Hashguch connecton. So, the queston s: whch one of these connectons s more sutable for the theory of Fnsler submanfolds? Secondly, most of the theores of Fnsler submanfolds have been developed by usng an nduced nonlnear connecton, whch n general does not concde wth the canoncal nonlnear connecton of the submanfold. Fnally, the structure equatons obtaned so far have cumbersome forms whch are almost mpossble to be used n a study of the geometry of a concrete Fnsler submanfold. Several people have made some fundamental contrbutons to the subject: Akbar Zadeh [1], Barthel [3], Bejancu [4], [5], Bejancu Farran [6], [7], Comc [8], Hamovc [9], Matsumoto [10], [11], Mron [13], Rund [14], Shen [15], Varga [17], Wegener [18]. The purpose of ths paper s to develop a theory of Fnsler submanfolds whch does not make use of the nduced nonlnear connecton and the nduced Fnsler connecton on a Fnsler submanfold F m = (M, F ) of F m+p = ( M, F ). To ths end, we ntroduce the normal curvature vector feld n of F m, and show that all structure 2010 Mathematcs Subject Classfcaton. Prmary 53C60; Secondary 53C40. Key words and phrases. Berwald connecton, Fnsler submanfold, flag curvature, normal curvature vector, Randers spheres, structure equatons, totally geodesc Fnsler submanfolds. 108

2 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 109 equatons of F m are smply expressed n terms of some covarant dervatves of n (cf. (3.7), (3.16) and (3.17)). Ths leads us to a smple relaton between the flag curvature of F m and F m+p (cf. (3.19)). The man tool n our study s the Berwald connecton on both F m and F m+p. Now, we outlne the content of the paper. In the frst secton we arrange some basc formulae from Fnsler geometry and recall the man propertes of vectoral Fnsler connectons (cf. Bejancu [5]). Next, n Secton 2 we relate the Berwald connectons on both F m and F m+p (cf. (2.23)). Then we ntroduce the normal curvature vector feld n and show that all nduced geometrc objects on F m are expressed n terms of the relatve vertcal covarant dervatves of n (cf. Theorem 2.1). In the last secton we obtan the structure equatons for a Fnsler mmerson, relatng the curvature tensor felds of Berwald connectons on F m and F m+p (cf. Theorem 3.2). Also, we relate the flag curvatures of F m and F m+p (cf. (3.19), (3.25)), and prove that a totally geodesc Fnsler submanfold has the same flag curvature as ts ambent Fnsler manfold. Fnally, we nvestgate the exstence of totally geodesc submanfolds of Randers spheres. The smple forms of the structure equatons for a Fnsler mmerson, whch are very rarely n Fnsler geometry, lead us to the concluson that the Berwald connecton s the best for a study of the geometry of Fnsler submanfolds. 1. Prelmnares Let M be an m-dmensonal manfold and T M be ts tangent bundle. Then we take (u, v ) as local coordnates on T M, where (u ) are the local coordnates on M and (v ) are the fber coordnates. Suppose that there exsts a functon F : T M [0, ), that s smooth on T M = T M \ {0} and satsfes the condtons: () F s postvely homogeneous of degree one wth respect to (v ), that s, F (u, kv) = kf (u, v), for any u M, v T u M, and k > 0. () The m m matrx [g β (u, v)], where we put g β (u, v) = 1 2 F 2 2 v v, β s a postve defnte quadratc form for all (u, v) T M. Then F m = (M, F ) s called an m-dmensonal Fnsler manfold. Denote by g β the entres of the nverse matrx of [g β ]. The functon F and the Fnsler tensor feld g = (g β ) are called the fundamental functon and the Fnsler metrc of F m, respectvely. In partcular, f F m s a Remannan manfold, then g becomes the Remannan metrc on M. As all the geometrc objects we work wth are supposed to be smooth (dfferentable of class C ), we must consder them defned on T M. However, to smplfy the notatons, from now on we wll omt the superscrpt from T M. Denote by F(T M) the algebra of smooth functons on T M and by Γ(E) the F(T M)-module of the smooth sectons of a vector bundle E over T M. Also, we use the Ensten conventon, that s, repeated ndces wth one upper ndex and one lower ndex denotes summaton over ther range. If not stated otherwse, we shall use the followng ranges for ndces: a, β,,... {1,..., m}, a, b, c,... {m + 1,..., m + p},, j, k,... {1,..., m + p}. Now, we consder the kernel V T M of the dfferental of the projecton mappng Π : T M M, whch s known as the vertcal bundle over T M. Locally, Γ(V T M)

3 110 AUREL BEJANCU AND HANI REDA FARRAN s spanned by {/v 1,..., /v m }. Then we consder the local vector felds (1.1) where we put (1.2) (a) G β = Gβ v, u = u Gβ (b) G β = 1 4 gβ v β, { 2 F 2 v u v F 2 } u It s proved that there exsts on T M a complementary dstrbuton HT M to V T M n T T M, whch s locally spanned by {/u 1,..., /u m }. We call HT M the canoncal horzontal dstrbuton on T M. In lterature, HT M s also known as the canoncal nonlnear connecton on T M. Now, we consder the decomposton and for any X Γ(T T M) we put T T M = HT M V T M, (1.3) X = hx + vx, where h and v are the projecton morphsms of T T M on HT M and V T M, respectvely. Also, ths decomposton of T T M enables us to defne the followng almost product structure on T M: (1.4) Q ( u ) = ( ) v, and Q v = u We call Q the Fnsler almost product structure on T M. Next, we consder a Fnsler connecton F C = (HT M, ), where HT M s the canoncal horzontal dstrbuton and s a lnear connecton on the vertcal bundle V T M. Then, we put u β v = F β v, and v β v = C β v, and deduce that, locally, a Fnsler connecton s determned by the trple (G, F β, C β ). In lterature there are four classcal Fnsler connectons: Berwald connecton, Cartan connecton, Chern Rund connecton, and Hashguch connecton. In the present paper we use only the Berwald connecton BF C = (HT M, ) = (G, G β, 0), that s, we have (1.5) (a) u β where we put v = G β v, (1.6) G β = G β v Denote by R the curvature tensor feld of and put ( ) (a) R u, u β v = H µ β (1.7) (b) (c) ( R v, ( R v, u β v β ) v = G µ β ) v = S µ β (b) v β v = 0, v µ, v µ, v µ

4 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 111 Then we recall that (cf. Bejancu Farran [7], p. 40, Matsumoto [12], pp. 118, 119) (1.8) where (a) (b) H µ β = G µ β u G µ µ µ u β + G β G G G β = Rµ β v, G µ β = G µ β v, (c) S µ β = 0, (1.9) R µ β = Gµ β u Gµ u β Also, we recall the followng Le bracket formulae: [ ] (1.10) (a) u, β u = R µ β v, µ (b) [ u β, ] µ v = G β v µ Due to (1.10a) we call R µ β the ntegrablty tensor feld of the canoncal horzontal dstrbuton HT M. The Cartan and Chern Rund connectons have F β gven by F β = 1 2 g { g u β Also, the Cartan tensor feld g β gven by + g β u g β u g β (1.11) g β = 1 2 v, has a great role n Fnsler geometry. It s mportant to note that the Berwald connecton s nether h-metrcal, nor v-metrcal connecton. More precsely, we have (1.12) (a) g β = 2g (F β G β ), (b) g β = 2g β, where and represent the horzontal and vertcal covarant dervatves wth respect to Berwald connecton. The homogenety of the fundamental functon mples some useful denttes: (1.13) (a) F β v = G β, (b) G β v = G β, (c) G β vβ = 2G, (d) g β v = 0, (e) H µ β v = R µ β, (f) G µ β v = 0. Then, by usng (1.12a), (1.13a) and (1.13b), we deduce that (1.14) g β v = 0. Now, we want to ntroduce the flag curvature of F m. Let (u, v) be a pont of T M, where u = (u ) s a pont of M and v = (v ) s a non-zero tangent vector to M at u. Suppose that X = (X ) s another tangent vector to M at u such that v and X are lnearly ndependent n T u M. Then, accordng to Bao Chern Shen [2], p. 68, we call the plane Π(X) = span{v, X} the flag at u wth flagpole v and transverse edge X. Then the flag curvature of F m at the pont u wth respect to the flag Π(X) s gven by (1.15) K(X) = H βv v β X X F 2 h X X, where H β = g µ H µ β, and h are the local components of the angular metrc gven by h = g l l, l = g µ v µ F }

5 112 AUREL BEJANCU AND HANI REDA FARRAN Next, we put (1.16) (a) R β = g µ R µ β, (b) R β v β = R, and note that R s a symmetrc Fnsler tensor feld. Then, by usng (1.13e), (1.16) and (1.15) we deduce that (1.17) K(X) = R X X F 2 h X X Denotng the denomnator n (1.17) by (X), t s easy to see that we have (1.18) (X) = g(x, X)g(v, v) g(v, X) Induced Geometrc Objects on a Fnsler Submanfold Let F m+p = ( M, F ) be an (m + p)-dmensonal Fnsler manfold and M be an m-dmensonal submanfold of M. Take (x, y ) and (u, v ) as local coordnates on T M and T M respectvely, and suppose that the mmerson of M n M s locally gven by the equatons x = x (u 1,..., u m ), {1,..., m + p}. To smplfy the equatons nvolved n the study, we make the notatons: B = x u, Bβ = 2 x u u, β Bβ 3 x = u u β u The fundamental functon F of F m+p nduces a functon on T M as follows: (2.1) F (u, v ) = F (x (u), y (u, v)), where we set (2.2) y (u, v) = B v. It s easy to check that F defnes a Fnsler structure on M. Then we say that F m = (M, F ) s a Fnsler submanfold of F m+p. The Fnsler metrcs g = (g β (u, v)) and g = ( g j (x, y)) are related by (2.3) g β (u, v) = g j (x(u), y(u, v))b B j β. Remark 2.1. The geometrc objects defned for F m n the prevous secton wll be consdered for F m+p, but wth a tlde, as for example: G, G β, R β become G k, G k j, R k j, respectvely. Next, by usng (2.2) we deduce that the natural feld of frames {/u, /v } and {/x, /y } on T M and T M, are related by (2.4) (a) u = B x + B βv β y, (b) v = B y Then we consder the Fnsler metrcs g = ( g j ) and g = (g β ) of F m+p and F m, as Remannan metrcs on the vertcal bundles V T M and V T M, respectvely. Due to (2.3) and (2.4b) we deduce that g s the nduced Remannan metrc on V T M by g. Ths enables us to consder the complementary orthogonal vector subbundle V T M to V T M n V T M restrcted to T M. Thus we have the orthogonal decomposton (2.5) V T M T M = V T M V T M. We call V T M the Fnsler normal bundle for the mmerson of F m nto F m+p.

6 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 113 Now, we choose a local feld of orthonormal frames wth respect to g n Γ(V T M ): Thus we have N a = N a y, a {m + 1,..., m + p}. (2.6) (a) g j BN a j = 0, (b) g j NaN j b = ab. [ ] B If s the nverse of the matrx [B Na], then we have: (2.7) Ñ a (a) B B β = β, (b) B N a = 0, (c) Ñ a B = 0, (d) Ñ a N b = a b, (e) B B j + N añ a j = j. By usng (2.3), (2.6) and (2.7e) we deduce that (2.8) (a) B = g j B j β gβ, (b) Ñ a j = g j N j b ba, (c) g β = g j B Bβ j. Also, the decomposton (2.5) s locally expressed as follows: (2.9) y = B v + Ñ a N a. Next, we consder the Cartan tensor felds of both F m and F m+p, and obtan (2.10) (a) g β = g jk B B j β Bk, (b) g β = g k j B B j β B k. Remark 2.2. Throughout the paper we shall use g β, g β, g j, g j, ab and ab for lowerng or rasng ndces for mxed Fnsler tensor felds wth local components T a... βjb.... We also need the followng mxed Fnsler tensor feld (2.11) g aβ = g jk N ab j B k β. Contractng (2.11) by Ñ a h and usng (2.7e), we obtan (2.12) g aβ Ñ a h = g hjk B j B k β g β B h. Now, from (2.8a) we deduce that g β Bβ = g j B j. Takng partal dervatves of ths equaton wth respect to v and usng (1.11) for Cartan tensor felds of both F m and F m+p, and (2.12), we nfer that (2.13) B β v = 2g a β Ñ a. Some basc denttes on a Fnsler manfold are stated n the next proposton. Proposton 2.1. Let F m be a Fnsler submanfold of F m+p. Then we have (2.14) G βb h + H a βn h a = B h βv + G h j B j β + D β B h, where we put (2.15) (a) H a β = Ñ a k (B k βv + G k j B j β ), (b) D β = g a β H a v.

7 114 AUREL BEJANCU AND HANI REDA FARRAN Proof. By usng (1.2b) for both F m and F m+p, and takng nto account (2.1), (2.4), (2.8), (2.7e) and (2.6a), we obtan (2.16) 2G = B k (2 G k + B k v v ). Then, takng the dervatves of ths equaton wth respect to v β and usng (1.2a) for both F m and F m+p, (2.13) and (2.4b), we deduced that (2.17) G β = g a β Ñ a k (2 G k + B k v v ) + B k ( G k j B j β + Bk βv ). On the other hand, takng nto account that G k are postvely homogeneous functons of degree two wth respect to y (see (1.9c)), and usng (2.2) we obtan (2.18) 2 G k = G k y j y j = G k j B j v. Hence, by usng (2.18) and (2.15) nto (2.17), we nfer that (2.19) G β = D β + B k ( G k j B j β + Bk βv ). Fnally, contractng (2.19) by B h and (2.15a) by N h a, and then addng the two equaltes we obtan (2.14) va (2.7e). Now, we are able to express {/u } gven by (1.1) n terms of {/x, /y }. Indeed, by usng (1.1) for both F m and F m+p, (2.4) and (2.14) we deduce that the canoncal horzontal dstrbuton HT M of F m s locally spanned by (2.20) u = B x + (Ha Na DB ) y Also, by usng (2.7b), (2.7d) and (2.13), we deduce that (2.21) B h v (Ha βna h ) = (Hβ an a h ) B h v = 2H a β g b (Ñ b h N h a ) = 2H a β g a. Next, let BF C = (HT M, ) = (G β, G β, 0) and BF C = (HT M, ) = ( G j, G j k, 0) be the Berwald connectons of F m and F m+p. Thus accordng to (1.5) we have (2.22) (a) x k Then we prove the followng. y j = G j k y, (b) y k y j = 0. Proposton 2.2. The local coeffcents G β and G j k of the Berwald connectons of F m and F m+p are related by (2.23) (G β + D β )B + H β b N b = B β + G j k B j β Bk, where we put (2.24) (a) D β = D β v 2Ha βg a, (b) H β b = Ñ b h v (Ha βn h a ).

8 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 115 Proof. Frst, we take the dervatves of (2.14) wth respect to v and by usng (1.6) for both F m and F m+p and (2.4b), we obtan ( ) G β D β v B h + v (Ha βna h ) = Bβ h + G h j k B j β Bk. Contractng ths n turn by B h and Ñ h b, and usng (2.7a), (2.21), (2.24), and (2.7c), we deduce that (2.25) (a) (b) G β + D β = B h (Bh β + G j h k B j β Bk ), H β b = Ñ b h (Bh β + G j h k B j β Bk ). Fnally, contractng (2.25a) by B and (2.25b) by Nb, and then addng, we obtan (2.23) va (2.7e). Now, we defne a tensor feld D on F m by ( ) (2.26) D v, v β = D β v, and a mxed Fnsler tensor feld H on F m wth respect to the Fnsler normal bundle V T M, as follows ( ) (2.27) H v, a v β = H β N a. Then, by usng (2.20), (2.4b), (2.22), (1.5), (2.26) and (2.27) we deduce that (2.23) s equvalent to (2.28) hx vy = hx vy + D(QhX, vy ) + H(QhX, vy ), for any X, Y Γ(T T M), where h and v are defned by (1.3) and Q s the Fnsler almost product structure on T M gven by (1.4). Accordng to the termnology from the theory of Remannan submanfolds, we call (2.28) (or, equvalently, (2.23)) the Gauss formulae for the Fnsler mmerson of F m n F m+p. Also, we note that the dfference hx vy hx vy s determned by a Fnsler tangent vector feld D(QhX, vy ) (.e., secton of V T M), and a Fnsler normal vector feld H(QhX, vy ) (.e., secton of V T M ). For ths reason we call D gven by (2.26) and (2.24a), and H gven by (2.27), (2.25b) and (2.24b), the tangent second fundamental form and the normal second fundamental form, respectvely. Moreover, we have the followng. Proposton 2.3. (2.29) () D β and H β a are symmetrc Fnsler tensor felds satsfyng the denttes (a) D β v = D β, (b) D β v β v = D β vβ = 0, (c) H β a v = H a β. () The Fnsler tensor felds D β and Dβ are postvely homogeneous of degrees 0 and 1 respectvely, wth respect to (v ).

9 116 AUREL BEJANCU AND HANI REDA FARRAN a Proof. () From (2.25) we see that both D β and H β are symmetrc Fnsler tensor felds wth respect to (β). Next, contractng (2.25a) by v and usng (1.13b) and (2.19) we obtan (2.29a). Smlarly, contractng (2.19) by v β and usng (1.13c), (2.16) and (2.29a) we deduce (2.29b). Fnally, (2.29c) s a consequence of (2.25b) and (2.15a). () By usng (2.13) we deduce that B are postvely homogeneous of degree 0. Then, from (2.25a), t follows that D β are postvely homogeneous of degree 0. Fnally, from (2.29a) we conclude that Dβ are postvely homogeneous of degree 1. Next, we make the notaton (2.30) H a 0 = H a βv β = H β a v β v, and defne on each coordnate neghbourhood of T M the secton n of V T M by (2.31) (a) n = H a 0 N a = n y, (b) n = H a 0 N a. Then t s easy to check that n defnes a global secton of V T M, that s, the local formula (2.31a) s nvarant wth respect to both the change of local coordnates on T M and the change of orthonormal bass n Γ(V T M ). As H a 0 /F 2, a {m + 1,..., m + p}, are called the normal curvatures of F m (cf. Bejancu [4]), we call n the normal curvature vector feld of F m. As we shall see n the next theorem, the normal curvature vector feld n determnes all the nduced geometrc objects of the Fnsler mmerson of F m nto F m+p. The followng notatons wll smplfy the presentaton: (2.32) (a) n = 1 n 2 v, (b) n β = 1 2 n 2 v v, β (c) n β = 1 3 n 2 v v β v Note that n, n β and n β are the local components of some mxed Fnsler tensor felds on F m wth respect to the vector bundle V T M T M (cf. Bejancu [5], p. 34). Now, we can state the man result of ths secton. Theorem 2.1. Let F m be a Fnsler submanfold of F m+p. Then the nduced geometrc objects H a β, D β, H β a, and D β are expressed n terms of n β and n β as follows: (2.33) (a) H a β = Ñ a n β, (b) D β = B n β, (c) H β a = Ñ a n β, (d) D β = B n β. Proof. Frst, takng nto account (2.32a), (2.31b), (2.7d), (2.24b) and (2.29c), we deduce that { HβN b b + v Ñ a n β = 1 2 Ñ a v β (v H b N b) = 1 2 Ñ a = 1 2 Ha β H β a v = H a β, } v β (Hb Nb) whch proves (2.33a). In a smlar way, by usng (2.32a), (2.31b), (2.7b), (2.21) and (2.15b) we obtan (2.33b). Next, contractng (2.33a) and (2.33b) by Na k and B k

10 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 117 respectvely, and subtractng these equatons, we nfer that (2.34) H a βn k a D β B k = n k β, va (2.7e). Then, takng the dervatves of (2.34) wth respect to v and by usng (2.7c) and (2.24b), we obtan Ñkn b k β = Ñ k b v (Ha βna k ) (Ñ kb b ) k D β v = H β b. Thus the proof of (2.33c) s done. Fnally, we take agan the dervatves of (2.34) wth respect to v and by usng (2.7a), (2.7b), (2.13), (2.7d) and (2.24a), we deduce that B k nk β = B k v (Ha βn k a ) D β v = 2Hβ an a k g b Ñk b D β v Ths completes the proof of the theorem. = (Ha βn k a ) B k v D β v = 2Ha βg a D β v = D β. Corollary 2.1. Let F m be a Fnsler submanfold of F m+p. Then we have: (2.35) G βb + n β = B βv + G jb j β, and (2.36) G β B + n β = B β + G j k B j β Bk. Proof. By usng (2.34) nto (2.14), we obtan (2.35). Then, contractng (2.33a) by N k a, and (2.33b) by B k, and then addng and usng (2.7e), we deduce that (2.37) H β a N k a + D β B k = n k β. Fnally, (2.36) s obtaned from (2.23) by usng (2.37). Corollary 2.2. Let F m be a Fnsler submanfold of F m+p. Then we have the assertons: () The canoncal horzontal dstrbuton of F m s locally spanned by the vector felds: (2.38) u = B x + n y () The local components of the normal curvature vector feld are postvely homogeneous of degree 2 wth respect to (v β ). Proof. By usng (2.20) and (2.34) we obtan (2.38). Then, contractng (2.34) by v β, and usng (2.32a) and (2.31b), we deduce that n k (2.39) v β vβ = 2n k. Thus, the asserton () follows from (2.39) by usng the Euler Theorem for postvely homogeneous functons. Next, for any X Γ(T T M) we defne the dfferental operator (2.40) X : Γ(V T M T M ) Γ(V T M T M ), X Ỹ = X Ỹ, Ỹ Γ(V T M T M ),

11 118 AUREL BEJANCU AND HANI REDA FARRAN where s the lnear connecton from the Berwald connecton BF C of F m+p gven by (2.22). Therefore BF C = (HT M, ) s a vectoral Fnsler connecton on F m wth respect to the vector bundle V T M T M (see Bejancu [5], p. 26). By (2.40) and (2.22) we deduce that s locally gven by (2.41) (a) u y j = G j y, (b) G j = G j k B k, (c) v y j = 0. Also, we note that B, {1,..., m} and {1,..., m+p} are the local components of a mxed Fnsler tensor feld on F m wth respect to the vector bundle V T M T M. Now, we can state the followng mportant result. Theorem 2.2. Let F m be a Fnsler submanfold of F m+p. Then the horzontal relatve covarant dervatve of B wth respect to the par (BF C, BF C) s completely determned by the normal curvature vector feld n of F m as follows: (2.42) B β = n β. Proof. By usng the local formulae for the horzontal relatve covarant dervatve (cf. Bejancu Farran [7], p. 29) and takng nto account (2.41b) and (2.36), we obtan = B β + Bj G j β G β B B β Thus the proof s done. = B β + G j k B j B k β G βb = n β. Fnally, by usng (2.41b), (2.38) and (1.11), we deduce that (2.43) g j = g j u g h h hjg g h G j ( gj = x k g G h h hj k g h Gj k = g j k B k + 2n k g jk. ) B k + n k g j y k 3. Structure Equatons and Curvature of a Fnsler Submanfold The purpose of ths secton s to obtan the structure equatons of the mmerson of F m n F m+p. More precsely, we want to relate R β wth R jk and {H β, G β } wth { H j kj, G j kh }. Ths wll enable us to relate the flag curvatures of F m and F m+p. Frst, we consder the assocate lnear connecton D to the Berwald connecton BF C = (HT M, ) = ( G j, G j k, 0) gven by (cf. Bejancu Farran [7], p. 30) (3.1) DX Y = X ṽy + Q X Q hy, X, Y Γ(T T M), where h and ṽ are the projecton morphsms of T T M on HT M and V T M respectvely, and Q s the Fnsler almost product structure on T M gven by ( ) (3.2) Q x = ( ) y, Q y = x

12 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 119 Then, by usng (3.1), (3.2) and (2.22), we obtan (3.3) (a) (c) D x k D y k x j = G j k x, (b) D x k x j = 0, (d) D y k y j = G j k y j = 0. Takng nto account (2.38), (3.3) and (2.41b) we deduce that (3.4) D u u β = (B β + G j k B j β Bk ) x + ( n β u + nj β G j y, ) y Next, we denote by T the torson tensor feld of D and by usng (3.4) and (1.10a) we nfer that ( ) [ ] T u, u β = D u u β D u β u u, u β {( ) ( )} n (3.5) β = u + nj β G j n n G β u β + nj G j β n G β y R β v = (R βb + n β n β ) y On the other hand, by usng (1.10) for F m+p and (3.3) we deduce that ( ) ( ) ( ) T x, k x j = R jk y and T x, k y j = T y, k y j = 0. Hence we have ( ) T u, u β (3.6) = T = T ( B k ( x k, x k + nk x j y k, B j β Comparng (3.5) and (3.6) we can state the followng. x j + nj β ) B j β Bk = R jkb j β Bk ) y j y Theorem 3.1. Let F m be a Fnsler submanfold of F m+p. Then we have: (3.7) R jk B j β Bk = R βb + n β n β, where stands for the horzontal relatve covarant dervatve wth respect to the par (BF C, BF C). Remark 3.1. By smlar calculatons for T (/u, /v β ) and T (/v, /v β ) we obtan (2.36) and a trval dentty, respectvely. (3.8) Next, by usng (2.38), (2.4b), (2.22) and (2.36) we obtan (a) (b) uβ vβ v = (B β + G j k B j B k β) y = (G βb + n β) y, v = 0.

13 120 AUREL BEJANCU AND HANI REDA FARRAN Takng nto account (2.38), (2.22) and the frst equalty n (3.8a) we deduce that u u β = ( B β + v = u (B β + G j k B j B k β) y u ( G j k B j B k β) + G j k B j β Bk + G j t k Gt h B j B k βb h ) y On the other hand, by drect calculatons usng (1.8b) for F m+p and (2.38) we obtan u ( G j k BB j β) k = G j x h k B j B k βb h + G j kh B j B k βn h + G j k B B k β + G j k B j B k β. Thus we have { ( u u β v = Bβ + G ) j k x h + G t j k Gt h BB j βb k h (3.9) } + G j kh BB j β knh + G j k BB β k + G j k BB j β k + G j k B j β Bk y Also, by usng (1.10a) and (3.8b) we nfer that (3.10) [ u, u β ] v = 0. Then, we denote by R the curvature tensor feld of the lnear connecton from the Berwald connecton BF C = (HT M, ) on F m+p, and by usng the frst equalty n (1.8a) for F m+p, (3.9) and (3.10) we obtan (3.11) R ( u, u β ) v = u u β v u β u v [ u, = { Hj kh B j B k β Bh + G j kh B j (B k β nh B k n h β u β ] }. v On the other hand, takng nto account the second equalty n (3.8a) and usng (2.41) and (2.36) we deduce that (3.12) u u β = = { G β u { G β u v = u (G βb + n β) y + (G βb + n β) u } B + G β B + n β u B + G β (B + G j k B j B k ) + n β u { (G β µ = u + G β G µ + G βb j G j k B k + n j β G j } + nj β G j y } ) B + n G β + n β u + nj β G j y y y

14 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 121 Fnally, by a smlar calculaton as n (3.11), but by usng (3.12), (3.10) and (1.8), we obtan ( ) (3.13) R u, { } u β v = H β B + n β n β y Thus, comparng (3.11) wth (3.13) we deduce that (3.14) Hj kh B j B k βb h + G j kh B j (B k βn h B k n h β) = H β B + n β n β. Next, by usng the frst equalty n (3.8a) and takng nto account (1.8b) for F m+p, (2.4b) and (2.41c) we nfer that v uβ v = G j kh B j B k βb h y On the other hand, by usng the second equalty n (3.8a) and takng nto account (1.8b), (2.41c) and (2.32c) we obtan v uβ v = (G βb + n β) y Comparng these equaltes we deduce that (3.15) Gj kh B B k βb h = G β B + n β. Now, we are able to state the followng. Theorem 3.2. Let F m be a Fnsler submanfold of F m+p. Then the curvature tensor felds of Berwald connectons on F m and F m+p are related by the followng structure equatons: (3.16) (a) Hjtkh B j B t µb k β Bh + G jtkh B j B t µ(b k β nh B k n h β ) = H µβ + g t (n β n β )Bt µ, and (3.17) (b) Gjtkh B j B t µb k β Bh = G µβ + g t n β Bt µ. (a) Hjtkh B j N t ab k β Bh + G jtkh B j N t a(b k β nh B k n h β ) = g t (n β n β )N t a, (b) Gjtkh B j N t ab k β Bh = g t n β N t a. Proof. By contractng (3.14) by g t B t µ and g t N t a we obtan (3.16a) and (3.17a). In a smlar way we obtan (3.16b) and (3.17b) from (3.15). Accordng to the termnology from the theory of Remannan submanfolds, we call (3.16) and (3.17) the Gauss Berwald equatons and the Codazz Berwald equatons for the mmerson of F m nto F m+p. Now, we want to relate the flag curvatures of F m and F m+p. To ths end we start wth a contracton of (3.7) by g h v β and by usng (1.16) for both F m and F m+p and (2.2) we obtan R hk B k = R µ Bµ h + g h(n β n β )vβ. Contractng ths by B h and usng (2.7a) we deduce that (3.18) Rhk B h B k = R + g h (n β n β )Bh v β.

15 122 AUREL BEJANCU AND HANI REDA FARRAN Theorem 3.3. Let F m = (M, F ) be a Fnsler submanfold of F m+p = ( M, F ), and X = (X ) be a tangent vector to M at the pont u M. Then the flag curvatures of F m and F m+p at the pont u wth respect to the flag Π(X) are related by (3.19) K(X) = K(X) + g h(n β n β )Bh v β X X F 2 h X X Proof. Contractng (3.18) by X X and settng X = B X, we deduce that (3.20) Rhk Xh Xk = R X X + g h (n β n β )vβ B h X X. If X = ( X ), then t s easy to check that g(x, X) = g( X, X), g(v, X) = g(y, X) and g(u, v) = g(y, y). Therefore, (X) from (1.18) s the same wth ( X) for F m+p. Then, dvdng (3.20) by (X) and takng nto account (1.17) for both F m and F m+p, we obtan (3.19). From (3.19) we see that the flag curvatures of F m and F m+p are related by means of the relatve horzontal covarant dervatves of n wth respect to the par (BF C, BF C). However, accordng to the theory of Remannan submanfolds, the second fundamental forms of F m (defned n Secton 2) should be nvolved n (3.19). To show ths, we examne the last term n (3.19). Frst, by usng (2.8a), (2.34), (2.7a) and (2.7b), we obtan g h n βb h = g B n β = g B (H a βn a D µ β B µ) = g D β = D β. Takng the relatve horzontal covarant dervatves of ths equalty, and usng (2.43), we obtan g h n β Bh = D β g h k n βb h B k 2 g hk n βb h n k g h n βb h. On the other hand, by usng (2.34), (2.37), (2.6) and (2.3), we nfer that g h n βb h = g h(h a βn a D µ β B µ)(h b N h b + D B h ) = H a H a β D D β. Thus we have g h n β Bh = D D β D β H a H a β g h k n βb h B k 2 g hk n βb h n k. By usng ths equaton and takng nto account that g hk s a symmetrc Fnsler tensor feld, we deduce that (3.21) g h (n β n β )Bh = D β D D D β + D β D β + H a H a β H aβh a + g h k (n β Bk n B k β )Bh. Next, by usng the last equalty n (2.29b) and takng nto account that v β = 0, we obtan (3.22) D β v β = 0. Also, we have (3.23) g h k B k βv β = g h k y k = 0, snce the horzontal covarant dervatve of g h wth respect to Berwald connecton n the drecton of the supportng element vanshes (cf. Matsumoto [12], p. 119).

16 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 123 Now, contractng (3.21) by v β and usng (3.22), (3.23), (2.29), (2.30) and (2.39), we deduce that (3.24) g h (n β n β )Bh v β = H0 a H a H a H a D β v β DD + 2 g h k n BB h. k Fnally, by usng (3.24) nto (3.19), we obtan K(X) = K(X) 1 + (3.25) F 2 h X X {Ha 0 H a H a H a D β v β DD + 2 g h k n BB h }X k X. We close the paper wth some applcatons of (3.25) to some specal Fnsler mmersons. Frst, we recall that F m s totally geodesc mmersed n F m+p f and only f any geodesc of F m s a geodesc of F m+p. To reach our goal, we need the followng. Theorem 3.4. (Bejancu [4]) F m s totally geodesc mmersed n F m+p, f and only f, the normal curvatures of F m vansh, that s, we have (3.26) H a 0 = 0, a {m + 1,..., m + p}. Ths enables us to state the followng. Theorem 3.5. F m s totally geodesc mmersed n F m+p, f and only f, both the normal and tangent second fundamental forms vansh, that s, we have (3.27) (a) H a β = 0, and (b) D β = 0,, β, {1,..., m}, a {m + 1,..., m + p}. Proof. Suppose that F m s a totally geodesc Fnsler submanfold. Then by (3.26) and (2.31b) we deduce that n = 0, for all {1,..., m + p}. Thus, takng nto account (2.33c), (2.33d) and (2.32b), we obtan (3.27). The converse s a smple consequence of (2.30) va (3.27a) and (3.26). Theorem 3.6. Let F m be a totally geodesc Fnsler submanfold of F m+p. Then F m and F m+p have the same flag curvature. Proof. As F m s totally geodesc, by (3.27a) and (2.29b) we deduce that (3.28) H a = 0, {1,..., m}, a {m + 1,..., m + p}. Also, by usng (3.27a) and (2.29a) we obtan (3.29) D β = 0,, β {1,..., m}. Fnally, by usng (3.28), (3.29), (3.26) and takng nto account that n = 0 for all {1,..., m + p}, from (3.25) we nfer that K(X) = K(X), X Γ(T M). Thus the proof s done. In partcular, we obtan an extenson to Fnsler geometry of a well known result from Remannan geometry. Corollary 3.1. Any totally geodesc Fnsler submanfold F m of a Fnsler manfold F m+p of constant flag curvature K s of constant flag curvature K, too.

17 124 AUREL BEJANCU AND HANI REDA FARRAN Next, we suppose that M s endowed wth a Remannan metrc ã = (ã j (x)) and a non-zero 1-form b = ( b (x)) satsfyng Then the functon b 2 = ã j (x) b (x) b j (x) < 1. (3.30) F (x, y) = ã j (x)y y j + b (x)y, defnes a Fnsler structure on M. Accordng to the termnology n lterature, we call F m+p = ( M, F, ã j, b ) a proper Randers manfold. By usng ã and b we defne the 1-form (3.31) θ = b ( b j b j )dx j, where denotes the covarant dervatve wth respect to the Lev Cvta connecton on ( M, ã). Now, we consder the (2n + 1) dmensonal unt sphere S 2n+1 and recall that Tanno [16] has proved that S 2n+1 s a Sasakan space form of constant ϕ-sectonal curvature c > 3. Then t was proved by Bejancu and Farran [6] that for any constant K > 0 there exsts a proper Randers metrc of constant flag curvature K and wth θ = 0 on S 2n+1. The Fnsler manfold F 2n+1 = (S 2n+1, c, K) s called a Randers (c, K)-sphere. The followng classfcaton theorem s useful n our study. Theorem 3.7. (Bejancu Farran [6]) Let F m+p = ( M, F, ã j, b ) be a proper Randers manfold, where ( M, ã) s a smply connected and complete Remannan manfold. Suppose that F m+p s of postve constant flag curvature K and that θ = 0 on M. Then m + p must be an odd number 2n + 1, and F 2n+1 s Fnsler sometrc to the Randers (c, K)-sphere F 2n+1 = (S 2n+1, c, K) where c = 1 4 b 2. Let F m = (M, F ) be a Fnsler submanfold of a proper Randers manfold F m+p = ( M, F, ã j, b ). Suppose that the structure vector feld b = ã j b j of F m+p s tangent to M. Then F m nherts a Randers structure gven by (3.32) (a) a β = ã j B B j β and (b) b = b B. Moreover, we prove the followng. Theorem 3.8. Let F m = (M, F ) be a totally geodesc Fnsler submanfold of a Randers (c, K)-sphere F 2n+1 = (S 2n+1, c, K), such that (M, a β ) s a smply connected and complete Remannan manfold, and the structure vector feld of F 2n+1 s tangent to M. Then m s an odd number 2q + 1 and F 2q+1 s Fnsler sometrc to a Randers (c, K)-sphere F 2q+1 = (S 2q+1, c, K). Proof. Frst, we note that F m s a proper Randers manfold wth the Remannan metrc and 1-form gven by (3.32). Then, by usng (3.32b) we deduce that (3.33) b β = B B j β b j, where the covarant dervatves are taken wth respect to the Lev Cvta connectons on the Remannan manfolds (M, a β ) and (S 2n+1, ã j ). By usng (3.33) and takng nto account that θ = 0 on S 2n+1, we deduce that θ = b (b β b β )du β = b B B j β ( b j b j )du β = b ( b j b j )dx j = 0.

18 THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION 125 Also, by Corollary 3.1 we nfer that F m s a Fnsler manfold of constant flag curvature K. Fnally, we apply Theorem 3.7 and obtan the asserton of the present theorem. The next corollary follows mmedately from Theorem 3.8. Corollary 3.2. There exst no totally geodesc even dmensonal Fnsler submanfolds of a Randers (c, K)-sphere F 2n+1 = (S 2n+1, c, K) whch are tangent to the structure vector feld of F 2n+1. Remark 3.2. We note that the Fnsler structure of a Randers (c, K)-sphere s never a Remannan structure. Ths s because b s nowhere zero on S 2n+1. Thus the results n Theorem 3.8 and Corollary 3.2 cannot be appled to Remannan geometry. References [1] Akbar Zadeh, H., Sur les sous-varétés des varétés fnslerennes, C.R. Acad. Sc. Pars, 266 (1968), [2] Bao, D., Chern, S.S. and Shen, Z., An Introducton to Remann-Fnsler Geometry, Graduate Text n Math., 200, Sprnger, Berln, [3] Barthel, W., Über de Mnmalflächen n gefaserten Fnslerräumen, Ann. d Mat., 36 (1954), [4] Bejancu, A., Specal mmersons of Fnsler spaces, Stud. Cercet. Mat., 39 (1987), [5] Bejancu, A., Fnsler Geometry and Applcatons, Ells Horwood, New York, [6] Bejancu, A. and Farran, H.R., On the classfcaton of Randers manfolds of constant curvature, Bull. Math. Soc. Sc. Math. Roumane, 52 (100), No. 3, 2009, [7] Bejancu, A. and Farran, H.R., Geometry of Pseudo-Fnsler Submanfolds, Kluwer Academc Publshers, Dordrecht, [8] Comc, I., The ntrnsc curvature tensors of a subspace n a Fnsler space, Tensor, N.S., 24 (1972), [9] Hamovc, M., Varétés totalement extrémales et varétés totalement géodésques dans les espaces de Fnsler, Ann. Sc. Unv. Jassy, 25 (1939), [10] Matsumoto, M., The nduced and ntrnsc Fnsler connectons of a hypersurface and Fnsleran projectve geometry, J. Math. Kyoto Unv., 25 (1985), [11] Matsumoto, M., Theory of Y -extremal and mnmal hypersurfaces n a Fnsler space, J. Math. Kyoto Unv., 26 (1986), [12] Matsumoto, M., Foundatons of Fnsler Geometry and Specal Fnsler Spaces, Kasesha Press, Sakawa, Ōtsu, [13] Mron, R., A non-standard theory of hypersurfaces n Fnsler spaces, An. St. Unv. Al.I. Cuza Ias, 30 (1974), [14] Rund, H., The Dfferental Geometry of Fnsler Spaces, Grundlehr. Math. Wss., 101, Sprnger, Berln, [15] Shen, Z., On Fnsler geometry of submanfolds, Math. Ann., 311 (1998), [16] Tanno, S., Sasakan manfolds wth constant ϕ-holomorphc sectonal curvature, Tôhoku Math. J., 21 (1969), [17] Varga, O., Über den nneren und nduzerten Zusammenhang für Hyperflächen n Fnslerschen Räumen, Publ. Math. Debrecen, 8 (1961), [18] Wegener, J.M., Hyperfächen n Fnslerschen Räumen als Transversalflächen ener Schar von Extremalen, Monatsh. Math. Phys., 44 (1936), Department of Mathematcs, and Computer Scence, Kuwat Unversty, P.O. Box 5969, Safat 13060, Kuwat. E-mal address: aurel.bejancu@ku.edu.kw, han.farran@ku.edu.kw