統計構造をもった接バンドルの共形射影平坦性について

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(1)Title. 統計構造をもった接バンドルの共形射影平坦性について. Author(s). 長谷川, 和泉; 中根, 敏幸; 奥山, 幸彦; 佐藤, 公威; 和田, 文興; 吉本, 拓郎. Citation. 北海道教育大学紀要. 自然科学編, 61(1): 41-56. Issue Date. 2010-08. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/2258. Rights. Hokkaido University of Education.

(2) ? k % B R B % b R W ( $ % F + b l ) %6lf % 13 Journal of Hokkaido University of Education (Natural Sciences) Vol. 61, No.1. %E22F 8 8 August, 2010. Remarks on Conformal-projective Flatness of Tangent Bundles with Some Lift Statistical Structures HASEGAWA Izumi , NAKANE ~ o s h i ~ u k iOKUYAMA *, ~ukihiko**,SAT0 ~imitake*** , WADA ~umioki****and YOSHIMOTO ~akuro***** Department of Mathematics, Sapporo Campus, Hokkaido University of Education, Sapporo 002-8502 *Sapporo Yamahana Junior High School. **Sapporo Nishino Junior High School M*. Hokkaido Otaru-ouyou High School. ****Hokkaido Sapporo Intercultural and Technological High School "*** Sapporo Ryukoku-gakuen High School. Abstract. Let (M, h, V) be a statistical manifold and T M the tangent bundle over M. If T M with "diagonal lift" statistical structure (hD, CD) is conformally-projectively flat, then (M, h, V) and (TM, hD, CD) are Hessian manifolds of constant Hessian curvature 0. If T M with statistical structure (hD, CH) is conformally-projectively flat, then (M, h, V) and (TM, hD, CH) are also Hessian manifolds of constant Hessian curvature 0.. 1 Introduction Let M be a differentiable manifold, h a semi-Riemannian metric on M, and V a symmetric affine connection on M. If the covariant derivative V h is symmetric, then V is said to be compatible t o h. A pair (h, V ) of semi-Riemannian metric h with compatible affine connection V is called a statistical structure on M. A manifold M together with a statistical structure (h, V ) is called a statistical manifold. A statistical manifold (M, h, V) is called a Hessian manifold if V is a flat connection..

(3) On a statistical manifold ( M ,h , V ) ,the symmetric tensor field C := V h is called the cubic form of M ;. where Skii := V k h j iand ( x h )is the local coordinate system of M . The tensor field S of type ( 1 , 2 ) is defined by. h ( S ( X ,Y ) ,2) := C ( X ,Y,2) = ( V xh )(Y,2). (1.2). for any X , Y,Z E r ( T M ) , where F ( E ) denotes the set of sections of a vector bundle E over M . S is called the skewness operator of ( M ,h , V ) ;. where. sjih:= sjiahahand (hj". := (hji)-'.. Then we have the following. 0. 0. where V denotes the Levi-Civita connection of h and 0. rji ( resp. (r)jih) are the coeffi-. cients of V (resp V ). For every statistical manifold ( M ,h , V ) ,there exists a naturally associated symmetric trilinear form C called the cubic form. Conversely let ( M ,h , C ) be a semi-Riemannian manifold with symmetric trilinear form C . Define a tensor field S of type (1, 2) by 0. h ( S ( X ,Y ) ,Z ) := C ( X ,Y,Z ) and an affine connection V by V := V - ; S . Then V is symmetric and satisfies V h = C . Hence the triplet ( M ,h , V ) becomes a statistical manifold. This symmetric affine connection V is called the compatible connection with respect to ( h , C ) . Thus equipping a statistical structure ( h ,V )is equivalent to equipping a pair ( h ,C) consisting of a semi-Riemannian metric h and a symmetric trilinear form C . Therefore ( h ,C ) is also called a statistical structure. In Riemannian geometry we have the following: Theorem A [15]. Let ( M , h ) be a semi-Riemannian manifold of dimension n ( 2 2 ) . Then ( T M , h C ) is conformally flat if and only if ( M , h ) is of constant curvature.. J. Inoguchi(cf. [ I l l ) suggested to generalize this theorem from the view point of statistical geometry. In our previous papers we proved the following: Theorem B [4]. Let ( M , h , V ) be a statistical manifold of dimension n ( 2 2 ) . Then ( T M , hC, V C )is conformally-projectively flat if and only if ( M , h , V ) is of constant curvature. Theorem C [5]. Let ( M , h , V ) be a statistical manifold of dimension n ( 2 2). If ( T M , h H , C H )is conformally-projectivelyflat, then ( M , h , V ) is of constant curvature..

(4) In this paper we prove the following: Theorem 1. Let (M, h, V) be a statistical manifold of dimension n ( 2 2). If (TM, hD, CD) is conformally-projectivelyfiat, then (M, h, V) and (TM, hD, CD) are Hessian manifolds of constant Hessian curvature 0.. Theorem 2. Let (M, h, V) be a statistical manifold of dimension n ( 2 2). If (TM, hD, CH) is conformally-projectivelyfiat, then (M, h, V) and (TM, hD, CH) are Hessian manifolds of constant Hessian curvature 0.. 2. Preliminaries. Let (M,h, V) be an n-dimensional statistical manifold. The curvature tensor K of (M, h, V) is defined by the curvature tensor of V, i.e.,. The curvature tensor satisfies Kkjih = - K .'Jkz. h1. K~~~~ + K~~~~ + Kzk'J Kkjih. = 0,. + Kjhik + Khkij = 0,. where Kkjih := Kkjiahah. The Ricci tensor of (M, h, V) is defined by. In statistical geometry, the Ricci tensor is not necessarily to be symmetric. The scalar curvature of (M, h, V) is defined by p := Rbahba. (M, h, V) is called an Einstein statistical manifold if p is constant and Rji = ihji. (M, h, V) is said to be of constant curvature if p is constant and Kkjih = -(6bhji 6:hki). (M, h, V) is said to be locally flat if V is a flat connection, i.e., K = 0. A locally flat statistical manifold (Ad,h, V) is also called a Hessian manifold.. Definition 2.1. A Hessian manifold (M, h, V) is said to be of constant Hessian c u r v a t u r e c if c is constant and. In [14], Shima called Q := ? V S E I'(TM('>~)) the Hessian curvature tensor, and defined the Hessian sectional curvature. * On a statistical manifold (M, h, V), the dual connection (or conjugate connection) V is introduced by. *. Then V is also compatible symmetric connection with respect to h and satisfies.

(5) and. * * * * where K denotes the curvature tensor of V and (K)kjih := (K)kjiahah.Accordingly, if * (h, V) is of constant curvature, then so is (h, V). In particular, if (h, V) is Hessian, then * so is (h, V). More generally, for any fixed real number a, we put. a. where. a. (r)jih denote the coefficients of 8 .. a. This symmetric affine connection V is called. a. 1. the a-connection (cf. [9]).Then (h, V) is a statistical structure on M . Note that V -1 * and V = V.. =. V. The following formulas are obtained by direct caluculation;. Using (2.5) we have. if (h, V) is a Hessian structure. Therefore we have the following: Lemma 2.2 [14]. If a Hessian manifold (M, h, V) is of constant Hessian curvature c, then h is of constant curvature - 2, that is,. Matsuzoe [lo] established the notion of conformal-projective equivalence relation and gave a necessary and sufficient condition for a conformally-projectively flat statistical manifold to be realized by a nondegenerate centroaffine immersion - of codimension 2. Definition 2.3. Two statistical structures (h, V) and (h, V) on M are said to be conformally-projectively equivalent if there exist two functions cp and $ on M satisfying:. -.

(6) and. 1.. e.,. A statistical manifold (M,h, V) is said to be conformally-projectively flat if (M, h, V) is conformally-projectively equivalent to a flat statistical manifold in a neighbourhood of an arbitrary point of M .. -. Remarks of Definition 2.3. (1) If cp = $, then (h, V) and (h, V) are conformally equivalent in statistical geometry.. equivalent. (2) If cp is constant, then V and ? are projectively (3) If $ is constant, then (h, V) and (h, V) are dual-projectively equivalent. Recently T. Kurose [8] defined a tensor field W, called the conformal-projective curvature tensor, as follows:. -. where R: := Rkahah,Q: := ~~~~~h~~ and Qji := Qjahai. This tensor W is invariant under conformal-projective change and plays the role in statistical geometry as same as the Weyl conformal curvature tensor does in Riemannian geometry. Proposition 2.4 [8]. A statistical manifold (M, h, V) of dimension n ( > 4 ) i s conformally-projectively flat if and only if the conformal-projective curvature tensor W vanishes everywhere o n M .. 3. Two statistical structures on TM. Let M be a differentiable manifold with an affine connection V, and rji the coefficients of V, i.e., Tji a& := V,di, where dh = and (xh) is the local coordinates of M. We define a local frame {Ei, E;) of T M as follows:. &. (3.1). E~:= ai - ybriba&. and. E; := d;,. 6.. where (xh, yh) is the induced coordinates of T M and d; := This frame {E,, E;) is called the adapted frame of T M with respect to V. Then {dxh, Syh) is the dual frame of {Ei, E;), where Syh := dyh ybrabhdxa.. +.

(7) Then, by straightforward calculation, we have the following Lemma 3.1. The Lie brackets of the adapted frame of T M satisfy the following identities: (1) [Ej, Ei1 = YbKQb"Ea, (2) [Ej, E;] = rji"Ea, (3) [ E j, E;] = 0) where K = ( K k j i h ) denotes the curvature tensor of V defined by Kkji := -. ajrki+ rji.rhh- rki. h. .. Let h be a semi-Riemannian metric with the coefficients hji on ( M ,V ) , i.e., h = hjidxj 8 dxi. The horizontal lift metric hH of h with respect to V is defined as follows:. The diagonal lift metric hD of h with respect to V is defined as follows:. Let M be a differentiable manifold with affine connection V . Let X = Xada be a vector field on M . The vertical lift XV and the horizontal lift X H of X are defined as follows:. (3.4). xV:= XaEa. and. xH:= XaEa.. We have the following lift tensors on T M ;. Let ( h ,V )be a statistical structure on M, and C the cubic form of ( h ,V ) .We consider , D )and ( h D C , H). two statistical structures ( h D C Lemma 3.2. Let V Dbe the compatible connection with respect to ( h D C , D )on T M . T h e n we have. We call this structure ( h D , V D )the diagonal lift statistical structure..

(8) Remark of Lemma 3.2. The usual diagonal lift connection [13]is the Levi-Civita connection of the diagonal lift metric hD. But this connection V D is different from the usual diagonal lift connection of V . This V Dis symmetric affine connection on T M and coincides with the usual diagonal lift connection if and only if S = 0. Lemma 3.3. the following:. For the diagonal lifi of h and the horizontal lifts of S and C , we have. Lemma 3.4. Then we have. Let. 7 be the compatible connection with respect to ( h D C , H )on T M . 1 2. 1 2. 1 2. O E j ~=. (rjia + -S,,")E~ - (-sjiU +-yb~jibu)~E, 1 1 1b u -(-sjiU + - y K j b i ) ~ u+ (rjiu + -s..~)E,, 2 2 2 1 1 1 V E j E i= - ( - S j i u + - y b ~ u i b j ) ~+ u -SjiuE,, 2 2 2 V E , ~=;. I V E - E ;= --S..aEu. 2. 4. Conformal-projective flatness of TM with ( h D ,C D ). Theorem 1. Let ( M , h, V ) be a statistical manifold of dimension n ( 2 2). If ( T M , h D , C D )is conformally-projectivelyfiat, then ( M , h, V ) and ( T M , hD, C D )are Hessian manifolds of constant Hessian curvature 0.. -. Proof. Suppose that ( T M , h D - , C D )is conformally-projectively flat. Then there exists a flat statistical structure ( h , V ) on T M which is conformally-projectively equivalent to ( h D ,C D ) .In this case we have. TA. where := on T M .. and. iA := TB(hD)BA( A ,B E { I , . - - , n , 1,0. -. .. , R ) ) for any function f.

(9) Let. k. be the curvature tensor of. ?. Using (4.1),we have (4.2),. - ., (4.7):. o = K ( E ~E, ~ ) E , =. (4.2). {K,,'. + 6,"al, - 6,"qi + hhA,". - hjiAt. +. -. (6,"hri- 6,"hji). 1 -ycYb 4 (Kjibd ~ a k c d- KkibdKajCd - 2KkjbdKaicd))Ea 1 hxiBja - hjiBka -ybviKkj; 2 1 - -4Y b ( s k f ~ , , - s j d a ~ k i b d 2sida~kjbd ~) )a .. +. +.

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(11) Since. and yji are defined by (4.8) and (4.10) respectively, we have olji = aij and We put Aji := Ajahai, Bji := Bjahai, Cji := Cjahai and Dji := Djahai. Then have also Aji = Aij, Bji = Cij and Dji = Dij. From (4.4)) we have aji. = yij.. Contracting (4.4)* with i and h, we have. Pkj= Cjk. From (4.7), we have. Contracting (4.7)* with i and h, we have Ckj= Cjk. Therefore, we obtain. Using (4.4)* and (4.15), we get. from which we have. Thus we have. from which. Applying the Ricci identities and using (4.16), we have. On the other hand, from (4.5) and (4.16), we get. From (4.17) and (4.18), we have Kkjih= 0. Therefore, from (4.16)*) and (4.18), (h, V ) is a Hessian structure of constant Hessian curvature 0. In this case, the curvature tensor KD of VD and VDSD also vanish..

(12) 5. Conformal-projective flatness of TM with ( h D ,CH). Theorem 2 . Let ( M , h, V ) be a statistical manzfold of dimension n ( 2 2). If ( T M , h D , C H )is conformally-projectively flat, then ( M , h, V ) and ( T M , hD, C H )are Hessian manifolds of constant Hessian structure.. Proof. Suppose that ( T M-, hD, - C H )is conformally-projectively flat. Then there exists a flat statistical structure ( h , V )on T M which is conformally-projectively equivalent to ( h D , C H ) .In this case we have. Let. k. be the curvature tensor of. ?. Using (5.1),we have (5.2),- - , (5.7):.

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(14) -. 0 = K ( E L ,Ej)Ei 1 =. (5.7). {,. 'sk;- skibs,jsa) + hkiCja- hjicka. (sji. -. 1 ,yb. :. ( s k i "KaCbj - Sji "KaCbk) }Ea. + { (ski'sjbasji'sk,")+ 6;yki 6;yji + hkiDja hjiD," + ($A@A) (6,"hki 6;hji)}Ea. -. -. -. -. -. Here we put as follows:. Since aji and yji are defined by (5.8) and (5.10) respectively, we have olji = aij and 73.2 . = yij. We put Aji := Ajahai,Bji := Bjahai,Cji := Cjahaiand D.. 11% := Djahai.Then we have also Aji = Aij, Bji = Cij and Dji = D,. From (5.5),we have.

(15) and. p..3% - p..23.. (5.16) From (5.7),we have. (5.17). Ckj= C j k .. From (5.4))we have. VkSjih = 2(Cjhhki- &hkh). 1. - -yb ( - 2VkKhibk. 2 from which we have. + SjhUKikbu- SkhaKiabj+ SjiuKkhbu - SkiUKuhbj) 1. VkSjih = (Cji - &)hkh + (Cjh - phj)hki = (Chi - a h ) hk) + (Chj - p j h ) hki = (Cij - hi)hkh + (Cih - phi) hkj Thus, using (5.16) and (5.17),we have (5.18). V k S j i h= 2 ( ~ i h pih) hy = 2(Chj - p h j ) hki Applying the Ricci identities and using (5.18),we have (5.19). = 2( ~ ji pji) hkh-. Kkjih= -Kkjhi.. From (5.15) and (5.19))we obtain Kkjih= 0. In this case we have the following:. (5.2)'. (5.7)'. B,. =0. and 6faki - 6kaji. Cji = 0 and 6:rki - 6;7ji Then, using these equations (5.2)*,(5.20). + hkiAj. - h, ,Ak. h + (JAgA)(6fhki- Jkhji) = 0,. + h x i ~ j-hhjio: + (JAgA)(6:hri - 6khji)= 0. , (5.7)*,we have. vks.!L 32 = S32 . . " ~ k u=hB.. JZ - C3% . . - 0,. a32.. - yji. and Aji = Dji.. Therefore ( h ,V ) is a Hessian structure of constant Hessian curvature 0. After simple cal, H )is also a Hessian structure of constant Hessian curvature culation we obtain that ( h DC 0 on TM..

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