GEOMETRIC INEQUALITIES FOR EINSTEIN TOTALLY REAL SUBMANIFOLDS IN A COMPLEX SPACE FORM

REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 58, No. 2, 2017, Pages 189 198 Published online: Mach 27, 2017 GEOMETRIC INEQUALITIES FOR EINSTEIN TOTALLY REAL SUBMANIFOLDS IN A COMPLEX SPACE FORM PAN ZHANG, LIANG ZHANG, AND MUKUT MANI TRIPATHI Abstact. Two geometic inequalities ae established fo Einstein totally eal submanifolds in a complex space fom. As immediate applications of these inequalities, some non-existence esults ae obtained. 1. Intoduction Accoding to Chen s conestone wok [1], the following poblem is fundamental: to establish simple elationships between the main intinsic invaiants and the main extinsic invaiants of Riemannian submanifolds. The basic elationships discoveed until now ae inequalities and the study of this topic has attacted a lot of attention duing the last two decades. Roughly speaking, thee ae thee main aspects of the study of this topic, one looking at the new Riemannian invaiants intoduced by Chen [2, 3, 4, 6, 10, 11, 17, 18, 20, 21, 23], the othe looking at the DDVV inequalities [7, 9, 14, 15, 16], and the last looking at the Casoati cuvatues [8, 12, 13, 19, 22]. In this pape, we ae inteested in obtaining chaacteizations of the elationships by Chen s invaiants. Let M be a Riemannian n-manifold and p a point in M. Suppose that K(π) is the sectional cuvatue of M with espect to a plane section π T p M. Fo each unit tangent vecto X of M at p, the Ricci cuvatue Ric(X) is defined by Ric(X) = n K(X e j ), j=2 whee {e 1, e 2,..., e n } is an othonomal basis of T p M with e 1 = X. In geneal, an n-dimensional manifold M whose Ricci tenso has an eigenvalue of multiplicity at least n 1 is called quasi-einstein. Fo instance, the Robetson Walke spacetimes ae quasi-einstein manifolds. Futhe, we say that M is an Einstein manifold if Ric(X) is independent of the choice of the unit vecto X. 2010 Mathematics Subject Classification. 53C40; 53C42. Key wods and phases. Inequalities; Einstein totally eal submanifolds; Complex space fom. The authos wee suppoted in pat by NSF in Anhui (No. 1608085MA03) and NSF fo Highe Education in Anhui (No. KJ2014A257). 189

190 P. ZHANG, L. ZHANG, AND M.M. TRIPATHI Then fo any unit tangent vecto X of M at p, one has Ric(X) = 2 n τ(p), whee τ(p) is the scala cuvatue at p defined by τ(p) = K(e i e j ). 1 i<j n Fo a given point p in M, let π 1,..., π q be q mutually othogonal plane sections in T p M, whee q is a positive intege n 2. Following [2], we define Kq inf K(π 1 ) + + K(π q ) (p) = inf, π 1 π q q whee π 1,..., π q un ove all mutually othogonal q plane sections in T p M. Fo each positive intege q n 2, define the invaiant δric q on M by δq Ric = sup Ric(X) 2q X Tp 1M n Kinf q (p), whee X uns ove all unit vectos in Tp 1 M := {X T p M : X = 1}. In [2], Chen established two inequalities in tems of the Riemannian invaiant δq Ric fo Einstein submanifolds in a eal space fom. As a natual polongation, in this pape, we obtain two inequalities fo Einstein totally eal submanifolds in a complex space fom. Unlike [2], we do not need the algebaic lemma fom [3]. Ou algebaic techniques also povide new appoaches to establish inequalities obtained in [2]. 2. Peliminaies Let N m be a complex m-dimensional Kähle manifold, i.e. N m is endowed with an almost complex stuctue J and with a J-Hemitian metic g. By a complex space fom N m (4c) we mean an m-dimensional Kähle manifold with constant holomophic sectional cuvatue 4c. A complete simply connected complex space fom N m (4c) is holomophically isometic to the complex Euclidean m-plane C m, the complex pojective m-space CP m (4c), o a complex hypebolic m-space CH m (4c) accoding to c = 0, c > 0 o c < 0, espectively. Denote by its Levi-Civita connection. The Riemannian cuvatue tenso field R with espect to has the expession R( X, Ỹ, Z, W ) = c ( X, Z Ỹ, W X, W Ỹ, Z + J X, Z JỸ, W J X, W JỸ, Z + 2 X, JỸ Z, J W ), fo any vecto fields X, Ỹ, Z, W on N m (4c). Let M be a totally eal submanifold in N m (4c). Accoding to the behavio of the tangent spaces unde the action of J, a submanifold M in N m (4c) is called totally eal if the complex stuctue J of N m (4c) caies each tangent space T p M of M into its coesponding nomal space T p M [5]. We denote the Levi-Civita connection of M by and by R the cuvatue tenso on M with espect to.

INEQUALITIES FOR EINSTEIN TOTALLY REAL SUBMANIFOLDS 191 The fomulas of Gauss and Weingaten ae given espectively by X Y = X Y + h(x, Y ), X ξ = A ξ X + Xξ, fo tangent vecto fields X and Y and nomal vecto field ξ, whee is the nomal connection and A is the shape opeato. The second fundamental fom h is elated to A ξ by h(x, Y ), ξ = A ξ X, Y. The mean cuvatue vecto H of M is defined by 1 H = tace h, n and we set H = H fo convenience. A submanifold M is called pseudo-umbilical if H is nonzeo and the shape opeato A H at H is popotional to the identity map. If H = 0, we say M is minimal. Besides, M is called totally geodesic if h = 0. Fo totally eal submanifolds, we have [5] XJY = J X Y, A JX Y = Jh(X, Y ) = A JY X. The above fomulas immediately imply that h(x, Y ), JZ is totally symmetic. Moeove, the Gauss equation is given by [5] R(X, Y, Z, W ) = c ( X, Z Y, W X, W Y, Z ) fo all vecto fields X, Y, Z, W on M. Choosing a local fame + h(x, Z), h(y, W ) h(x, W ), h(y, Z) e 1,..., e n, e n+1,..., e m, e m+1 = J(e 1 ),..., e m+n = J(e n ), e m+n+1 = J(e n+1 ),..., e 2m = J(e m ) in N m (4c) in such a way that, esticted to M, e 1, e 2,..., e n ae tangent to M. With espect to the local fame of N m (4c) chosen above, we denote the coefficients of the second fundamental fom h by {h ij }, 1 i < j n; n + 1 2m. 3. The fist inequality Theoem 3.1. Fo any intege k 2, let M be a 2k-dimensional Einstein totally eal submanifold of an m-dimensional complex space fom N m (4c) of constant holomophic sectional cuvatue 4c. Then we have δ Ric k 2(k 1)(c + H 2 ). (3.1) The equality case of (3.1) holds if and only if one of the following two cases occus: (i) M is a minimal and Einstein totally eal submanifold, such that, with espect to suitable othonomal fames {e 1,..., e 2k, e 2k+1,..., e 2m }, the shape opeatos of

192 P. ZHANG, L. ZHANG, AND M.M. TRIPATHI M take the following fom: A 1... 0 A =......., = 2k + 1,..., 2m, 0... A k whee A i, i = 1,..., k, ae symmetic 2 2 submatices satisfying tace(a 1) = = tace(a k ) = 0. (ii) M is a pseudo-umbilical and Einstein totally eal submanifold, such that, with espect to suitable othonomal fames {e 1,..., e 2k, e 2k+1,..., e 2m }, the shape opeatos of M take the following fom: A = A 1... 0..... 0... A k, = 2k + 2,..., 2m, whee A i, i = 1,..., k, ae symmetic 2 2 submatices satisfying tace(a 1) = = tace(a k ) = 0. Poof. Fo a given point p in M, let π 1,..., π k be k mutually othogonal plane sections at p. We choose an othonomal basis {e 1,..., e 2k } of T p M such that π 1 = Span{e 1, e 2 },..., π k = Span{e 2k 1, e 2k }. Since M is a 2k-dimensional Einstein manifold, we have τ = k Ric(X). Fom the definition of δk Ric and the Gauss equation, we have kδ Ric k = τ [K(π 1 ) + K(π 2 ) + + K(π k )] = k(2k 1)c + [h iih jj (h ij) 2 ] + + c + 2k(k 1)c + 1 i<j 2k [h 2k 1,2k 1h 2k,2k (h 2k 1,2k) 2 ] [ 1 i<j 2k 2k { c + } [h 11h 22 (h 12) 2 ] h iih jj (h 11h 22 + + h 2k 1,2k 1h 2k,2k) ] = 2k(k 1)c + 1 { ( h 2 ii) 2 [(h 11 + h 22) 2 +... i=1 } + (h 2k 1,2k 1 + h 2k,2k) 2 ] Using the Cauchy inequality, we obtain that (3.2) (h 11 + h 22) 2 + + (h 2k 1,2k 1 + h 2k,2k) 2 1 2k k ( h ii) 2, (3.3) with the equality case of (3.3) holds if and only if h 11 + h 22 = = h 2k 1,2k 1 + h 2k,2k. i=1

INEQUALITIES FOR EINSTEIN TOTALLY REAL SUBMANIFOLDS 193 Plunging (3.3) into (3.2), we have kδ Ric k 2k(k 1)c + 1 2 = 2k(k 1)c + k 1 2k { 2k ( h ii) 2 1 2k k ( h ii) 2} i=1 2k ( h ii) 2 i=1 = 2k(k 1)c + k 1 2k 4k2 H 2 = 2k(k 1)(c + H 2 ), which implies δk Ric 2(k 1)(c + H 2 ). Next, we will discuss the equality case. The equality case of (3.1) at a point p M holds if and only if we have the equality in (3.2) and (3.3), i.e. with espect to suitable othonomal fames, the shape opeatos take the following fom: A 1... 0 A =....., = 2k + 1,..., 2m, 0... A k whee A i, i = 1,..., k, ae symmetic 2 2 submatices satisfying tace(a 1) = = tace(a k). The est of the discussion is simila to that of the poof of Theoem 1 in [2]. 4. The second inequality Theoem 4.1. Let M be an n-dimensional Einstein totally eal submanifold of an m-dimensional complex space fom N m (4c). Then fo evey positive intege q < n 2, we have δ Ric q ( n 1 2q ) c + n i=1 n(n q 1) H 2. (4.1) n q The equality case of (4.1) holds if and only if M is a totally geodesic submanifold. Poof. Given a point p in M and a positive intege q < n 2, let π 1,..., π q be q mutually othogonal plane sections of M at p. We choose an othonomal basis of T p M such that Then fom the definition of δ Ric q π 1 = Span{e 1, e 2 },..., π q = Span{e 2q 1, e 2q }. we have nδ Ric q (p) = n Ric(X) 2[K(π 1 ) + + K(π q )] = 2q Ric(X) 2[K(π 1 ) + + K(π q )] + (n 2q) Ric(X). Fo convenience, we set I = 2q Ric(X) 2[K(π 1 ) + + K(π q )], II = (n 2q) Ric(X). (4.2)

194 P. ZHANG, L. ZHANG, AND M.M. TRIPATHI Now we compute I and II sepaately. Fist, we ewite I as q I = [Ric(e 2l 1, e 2l 1 ) + Ric(e 2l, e 2l ) 2K(π l )], l=1 which togethe with the Gauss equation gives I 2q(n 2)c + [ ( h 11h jj + h 22h jj + + h 2q,2qh jj) j 1 j 2 j 2q 2(h 11h 22 + h 33h 44 + + h 2q 1,2q 1h 2q,2q) ] = 2q(n 2)c + [ h iih jj + 2 h iih jj 1 i 2q, 2q+1 j n 1 i<j 2q 2(h 11h 22 + h 33h 44 + + h 2q 1,2q 1h 2q,2q) ] = 2q(n 2)c + { h iih jj + (h 11 + + h 2q,2q) 2 1 i 2q, 2q+1 j n } [(h 11 + h 22) 2 + + (h 2q 1,2q 1 + h 2q,2q) 2 ]. (4.3) On the othe hand, we can ewite II as II = Ric(e 2q+1, e 2q+1 ) + Ric(e 2q+2, e 2q+2 ) + + Ric(e n, e n ), which togethe with the Gauss equation gives II = (n 2q)(n 1)c + [h 2q+1,2q+1h jj (h 2q+1,j) 2 ] j 2q+1 + + [h nnh jj (h nj) 2 ] j n (n 2q)(n 1)c + ( h 2q+1,2q+1h jj + + j 2q+1 j n = (n 2q)(n 1)c + (2 h iih jj + 2q+1 i<j n h nnh jj) 1 i 2q, 2q+1 j n h iih jj). (4.4)

INEQUALITIES FOR EINSTEIN TOTALLY REAL SUBMANIFOLDS 195 Plunging (4.3) and (4.4) into (4.2), we obtain that nδq Ric (p) (n 2 n 2q)c + (h 11 + + h 2q,2q) 2 + + (h 2q 1,2q 1 + h 2q,2q) 2 ] + 2 + 2 h iih jj 1 i 2q, 2q+1 j n = (n 2 n 2q)c + 2q+1 i<j n (h 11 + + h 2q,2q) 2 + + (h 2q 1,2q 1 + h 2q,2q) 2 ] + 2 + [n 2 H 2 (h 11 + + h 2q,2q) 2 2q+1 i<j n [(h 11 + h 22) 2 h iih jj [(h 11 + h 22) 2 h iih jj (h 2q+1,2q+1 + + h nn) 2 ] (4.5) = (n 2 n 2q)c + n 2 H 2 [ (h 11 + h 22) 2 + (h 33 + h 44) 2 + + (h 2q 1,2q 1 + h 2q,2q) 2 + (h 2q+1,2q+1) 2 + + (h nn) 2]. Fom the Cauchy inequality, we know that (h 11 + h 22) 2 + + (h 2q 1,2q 1 + h 2q,2q) 2 + (h 2q+1,2q+1) 2 + + (h nn) 2 with the equality case of (4.6) holds if and only if 1 n q (h 11 + h 22 + + h nn) 2, (4.6) h 11 + h 22 = = h 2q 1,2q 1 + h 2q,2q = h 2q+1,2q+1 = = h nn. Then we plunge (4.6) into (4.5), namely, nδ Ric q (p) (n 2 n 2q)c + n 2 H 2 1 n q = (n 2 n 2q)c + n2 (n q 1) H 2, n q (h 11 + h 22 + + h nn) 2 which means δ Ric q (n 1 2q n n(n q 1) )c + H 2. n q Next, we will discuss the equality case. The equality case of (4.1) at a point p M holds if and only if we have the equality in (4.3), (4.4) and (4.6), i.e. with espect to suitable othonomal fames, the shape opeatos take the following fom:

196 P. ZHANG, L. ZHANG, AND M.M. TRIPATHI A 1... 0 0. A =....., = n + 1,..., 2m, 0... A k 0 0... 0 µ E whee E is the (n 2q) (n 2q) identity matix and A i, i = 1,..., k, ae symmetic 2 2 submatices satisfying tace(a 1) = = tace(a k) = µ. The est of the discussion is simila to that of the poof of Theoem 2 in [2]. 5. Immediate applications Fom Theoems 3.1 and 4.1 we obtain immediately the following. Coollay 5.1. If a Riemannian n-manifold M admits a totally eal isometic immesion into a complex Euclidean space which satisfies fo some positive intege q n 2 δ Ric q > n(n q 1) H 2, n q Theoems 3.1 and 4.1 also imply the following. at some point, then M is not an Einstein manifold. Coollay 5.2. If an Einstein n-manifold satisfies δq Ric > (n 1 2q n )c, fo some positive intege q n 2 at some point, then it admits no totally eal minimal isometic immesion into a complex space fom of constant holomophic sectional cuvatue 4c egadless of codimension. Besides, fom Theoems 3.1 and 4.1, we can also get Coollay 3 in [2]. 6. Acknowledgements The authos would like to thank Pofesso B. Y. Chen fo the discussions held on this topic. Refeences [1] B.-Y. Chen, Mean cuvatue and shape opeato of isometic immesions in eal-space-foms, Glasgow Math. J. 38 (1996), 87 97. MR 1373963. [2] B.-Y. Chen, A Riemannian invaiant and its applications to Einstein manifolds, Bull. Austal. Math. Soc. 70 (2004), 55 65. MR 2079360. [3] B.-Y. Chen, A geneal optimal inequality fo waped poducts in complex pojective spaces and its applications, Poc. Japan Acad. Se. A Math. Sci. 79 (2003), 89 94. MR 1976363. [4] B.-Y. Chen, A tou though δ-invaiants: Fom Nash s embedding theoem to ideal immesions, best ways of living and beyond, Publ. Inst. Math. (Beogad) (N.S.) 94(108) (2013), 67 80. MR 3137491.

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198 P. ZHANG, L. ZHANG, AND M.M. TRIPATHI P. Zhang Key Laboatoy of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, Univesity of Science and Technology of China, Anhui 230026, People s Republic of China panzhang@mail.ustc.edu.cn L. Zhang School of Mathematics and Compute Science, Anhui Nomal Univesity, Anhui 241000, P. R. China zhliang43@163.com M.M. Tipathi Depatment of Mathematics and DST-CIMS, Institute of Science, Banaas Hindu Univesity, Vaanasi 221005, India mmtipathi66@yahoo.com Received: Mach 30, 2016 Accepted: Novembe 8, 2016