《芬斯勒射影几何中的Ricci曲率_英文_》由会员分享，可在线阅读，更多相关《芬斯勒射影几何中的Ricci曲率_英文_（7页珍藏版）》请在金锄头文库上搜索。
1、J . of Mat h . ( P RC)THE RICCI CURVATURE IN FINSL ER3P ROJ ECTIVE GEOMETRYYA N G We n2mao (杨文茂) 1 , C H EN Xi n2yue (程新跃) 2( 1 . D e p t. o f M at h . , Y a n ge n U ni v e rs i t y , Q u a n z hou 362014 , Ch i n a)( S chool of M at h . an d Ph y . , Chon gqi n g I nstit ute of Technolog y , Chon
2、gqi n g 400050 , Chi na)Abstract : In t hi s p ap er we fir st st udy t he Finsler p rojective change which p reserves t he Riccicurvat ure . Furt her mo re ,given a co mpact a nd bo unda r yless n2dimensio nal diff erentia ble ma nifoldM ,we sho w t hat a ny point wi se C2p rojective cha nge f ro m
3、 a Berwald space ( M , F) to a Riema nnsp ace ( M , F) i s t rivial if t he t race of t he Ricci curvat ure Ric of F wit h resp ect to F i s less o r equal to t he scala r curvat ure of F.Key words : Finsler met ric ; geo de sic ; p rojective cha nge ; Ricci curvat ure ; scalar curvat ure2000 M R Su
4、bject Classif ication : 53C60 , 53C40Document code : AArticle ID : 025527797( 2005) 05204732071Introduct ionFi n sler p rojective geo met r y i s a n i mpo r t a nt p a r t of Fi n sle r geo met r y. Give n t wo Fi n sler met ric s F a nd F o n a n n2di me n sio nal ma nifol d M . We sa y F a nd F t
5、o be poi nt wi se p rojectivel y relat ed (o r t he cha nge F F i s a p rojective c ha nge) if a ny geo de sic of F i s al soa geo de sic of F a nd t he i nve r se i s al so t r ue . Two re gula r Fi n sler met ric sp ace s a re sai d to be p rojectivel y relat e d if t he re i s a diff eo mo rp hi
6、sm bet wee n t he m suc h t hat t he p ull2bac k met ric i s poi nt wi se p rojective to a no t her o ne . In ge neral , give n a Fi n sle r met ric F o n a ma nifol d M , we wo ul d li ke to det e r mi ne all Fi n sler met ric s o n M w hich a re poi nt wi se p rojective to F. Pa r ticula rl y , it
7、 i s i nt ere sti ng a nd mea ni ngf ul to det er mi ne all Fi n sler met ric s o n M w hich a re poi nt wi se p rojective to a Rie ma nn met ric o n M . The p ro ble m of cha ract e rizi ng a nd st udyi ng Fi n sle r met ric s w hic h a re poi nt wi se p rojective to a give n Fi n sler met ric i s
8、k no w n a s Hil be r t s fo ur t h p ro ble m . In 6 , Z. She n st udie d t he follo wi ng p ro ble m : give n a n Ei n st ei n met ric , de scri be all Ei n st ei n met ric s w hic h a re poi nt wi se p rojective to t he give n o ne . She ns re sult s a re ver y i nt e re sti ng. The fi r st a ut
9、ho r a nd Z.Received date : 20042022103Accepted October 30 , 2004 Foundation item : Suppo rt ed by t he Natio nal Nat ural Science Fundatio n of Chi na ( 10371138 ) ; t he Science Fo undatio n of Cho ngqi ng Educatio n Co mmit t eeShen st udied t he co mpariso n t heorem o n t he Ricci curvat ure in
10、 p rojective geo met ry 3 and discussed t he p rojectively flat Finsler met rics wit h almo st isot ropic S2curvat ure 3 . The Fi n sle r met ric s co n si dered i n t hi s p ap e r a re po sitive defi nit e Fi n sle r met ric s. We fi r st st udy t he Fi n sle r p rojective cha nge s w hich p re se
11、 r ve t he Ricci c ur vat ure . We p ro ve t hat : suppo se t hat ( M , F) a nd ( M , F) a re poi nt wi se p rojectivel y relat ed a nd t he p rojective change p reserves t he Ricci curvat ure ,t hen alo ng any geodesic c ( t) of F or F , F ( c ( t) ) / F ( c ( t) ) = co nstant if F is co mplete (cf
12、 . Theorem 4. 3) . Seco ndly ,we co nsider t he existence of p rojective changes between a Berwald met ric and a Riemann met ric o n manifold M. A ssume t hat M is a co mpact and bo undaryless n2dimensio nal differentiable manifold. We p rove t hat t here is not no nt rivial pointwise C2p rojective
13、change f ro m a Berwald space ( M , F) to a Riemann space ( M , F) ift rF Ric sF ( cf .Theorem 5. 1) . In particular , when ( M , F) and ( M , F) are bot h Riemann spaces ,we show t hat t here is not no nt rivial p rojective change f ro m ( M , F) to ( M , F) if t rF Ric sF (cf . Theorem 5. 2) . Ack
14、nowledgements.The aut hor wishes to exp ress his sincere t hanks to Prof . S. S. Chern and Prof . Z. Shen for t heir kind help and enco uragement .2Prel iminariesIn t his sectio n we shall give so me definitio ns in Finsler geo met ry. Relevant terminolo gies and notatio ns are referred to 1 2 o r 5
15、 6 . Given a Finsler space ( M , F) . The geo desic is given by x + 2 Gi ( x) = 0 ,1 gil (5l 5k F2 ) yk - 5l F2 and 5l = 5/ 5 yl , 5k = 5/ 5 x k .where Gi =Define Ry :Tx M Tx M by4 55Riv kv kRy ( v) =yk ( )( 1)| x ,v =| x ,5 x i5 x kiiGlGk-Ri ( y) =y l ,w here( 2)kx kx l 5 G 5 i l 5 Gi l = 5 y l , = - Gk .x k5 x k5 y lRy i s a well2defi ned li nea r t ra n sfo r matio n sati sf yi ng Ry ( y ) cur vat ure . Put= 0 .We call R t he Rie ma n niiGlGkRi kl = x k( 3)- x l ,i5 R klRi jkl =( 4)( 5) ( 6)5 y j .Riklj + Ri l kCl
