TEZĂ DE ABILITARE. Subvarietăți de curbură medie paralelă și subvarietăți biarmonice în varietăți riemanniene

ACADEMIA ROMÂNĂ SCOSAAR TEZĂ DE ABILITARE Subvarietăți de curbură medie paralelă și subvarietăți biarmonice în varietăți riemanniene Dorel Fetcu Domeniul fundamental Matematică și științe ale naturii Domeniul de abilitare Matematică Teză elaborată în vederea obţinerii atestatului de abilitare în scopul conducerii lucrărilor de doctorat în domeniul Matematică BUCUREŞTI, 2016

ACADEMIA ROMÂNĂ SCOSAAR HABILITATION THESIS Submanifolds with Parallel Mean Curvature and Biharmonic Submanifolds in Riemannian Manifolds Dorel Fetcu BUCHAREST, 2016

To Ilinca, Iuria, and Petronela

Abstract A classical but still very dynamic topic in the field of Differential Geometry is the study of surfaces with constant mean curvature (cmc surfaces) in 3-dimensional spaces and, more generally, of submanifolds with parallel mean curvature vector field (pmc submanifolds) in Riemannian manifolds with arbitrary dimension. Their history is spread on more than six decades, in the case of cmc surfaces, and goes back to the early 1970s, in the case of pmc submanifolds, to papers like [35] by B.-Y. Chen and G. D. Ludden, [52, 53] by J. Erbacher, [58] by D. Ferus, [83] by D. A. Hoffman, or [136] by S.-T. Yau. In the following, we shall briefly recall only some of those results that represent a source of inspiration for our work. Two very powerful tools were mainly used to prove these results: holomorphic differentials defined on cmc (or pmc) surfaces and Simons type equations. H. Hopf [85] was the first to use a holomorphic differential to show that any cmc surface homeomorphic to a sphere in Euclidean 3-space is actually a round sphere. His result was extended to cmc surfaces in 3-dimensional space forms by S.-S. Chern [41], and then to cmc surfaces in product spaces of type M 2 (c) R, where M 2 (c) is a complete simply-connected surface with constant curvature c, as well as Nil(3) and PSL(2, R), by U. Abresch and H. Rosenberg [1, 2]. The next natural step was to study pmc surfaces in product spaces of type M n (c) R, where M n (c) is a space form with constant sectional curvature c, i.e., those surfaces satisfying H = 0, where is the connection in the normal bundle and H is the mean curvature vector field. Two very important papers on this topic are [5, 6] by H. Alencar, M. do Carmo, and R. Tribuzy. In these articles they introduce a holomorphic differential (that generalizes the Abresch-Rosenberg differential defined in [1] for cmc surfaces in M 2 (c) R) and use it to study the geometry of pmc surfaces. One of the main results in [6] is a reduction of codimension theorem, showing that a pmc surface immersed in M n (c) R either is a minimal surface in a totally umbilical hypersurface of M n (c); or a cmc surface in a 3-dimensional totally umbilical or totally geodesic submanifold of M n (c); or it lies in M 4 (c) R. Another very effective method developed in order to study minimal or, more generally, cmc and pmc submanifolds in Riemannian manifolds, is to use Simons type equations. In 1968, J. Simons [128] discovered a fundamental formula for the Laplacian of the second fundamental form of a minimal submanifold in a Riemannian manifold and used it to characterize certain minimal submanifolds of a sphere and Euclidean space. One year later, K. Nomizu and B. Smyth [112] generalized Simons equation in the case of cmc hypersurfaces in a space form and their result was then extended, in B. Smyth s work [129], to the more general case of pmc submanifolds in a space form. Over the years such formulas, nowadays called Simons type equations, were i

ii used more and more often in studies on cmc and pmc submanifolds (see, for example, [4, 7, 8, 16, 18, 28, 39, 123]). During the last three decades one could observe an ever growing interest in the study of certain fourth order partial differential equations, which generalize the notion of harmonic maps. In their seminal paper [49], J. Eells and J. H. Sampson suggested the notion of biharmonic maps ψ : M N between two Riemannian manifolds, defined as critical points of the bienergy functional E 2 (ψ) = τ(ψ) 2 dv, M where τ(ψ) = trace ψ dψ is the tension field of ψ, ψ being the connection in the pull-back bundle ψ 1 T M. If M is not a compact manifold, then biharmonic maps ψ : M N are defined as solutions of the Euler-Lagrange equation τ 2 (ψ) = 0, where τ 2 (ψ) = τ(ψ) trace R(dψ, τ(ψ))dψ is the bitension field of ψ. It is easy to see that any harmonic map is biharmonic and that is why we are interested in proper-biharmonic maps, i.e., those biharmonic maps which are not harmonic. A special case is that of biharmonic Riemannian immersions, or biharmonic submanifolds, i.e., those submanifolds for which the inclusion map is biharmonic. This definition of biharmonic submanifolds coincides, when working in Euclidean space (and only then), with that proposed by B.-Y. Chen [31], where a biharmonic submanifold is characterized by the fact that its mean curvature vector field is harmonic. Although only non-existence results for proper-biharmonic submanifolds in Euclidean space were obtained (see, for example, [34, 48, 91, 92, 106]), when the ambient space is not flat, numerous examples and classification results for properbiharmonic submanifolds were found in papers like [12]-[14], [21]-[26], [44, 88, 100, 101, 104, 106], [117]-[119], [124]-[126], and [139]. Our thesis is organized in two parts, the first one, that contains two chapters, being devoted to the study of pmc submanifolds, while in the second part, consisting of three chapters, we consider biharmonic submanifolds. In the first chapter, we present results from [62], [77], and [78] on pmc surfaces in complex, cosymplectic, and Sasakian space forms, respectively. In all these situations, we prove reduction of codimension theorems and also introduce holomorphic differentials that are then used to study the geometry of some of these surfaces. The second chapter is devoted to the study of pmc submanifolds, this time of arbitrary dimension, in M n (c) R. We first prove two Simons type equations that are then used to obtain gap theorems for such submanifolds. We also consider pmc surfaces with finite total curvature and find a result on their compactness. The chapter ends with a classification result for helix pmc surfaces. These results were obtained in [17], [63], [72], [74], [75], and [76]. We begin dealing with biharmonic submanifolds in the third chapter, that is based on [71] and [72]. Here, we first present a gap theorem for pmc properbiharmonic submanifolds in M n (c) R and also classify pmc proper-biharmonic surfaces in this space. We then turn our attention to biconservative surfaces in M n (c) R, i.e., those surfaces for which the tangent part of the bitension field vanishes (we note that biconservative submanifolds have only very recently begun to be studied in articles like [27, 80, 107, 108]). We completely determine such

surfaces that have parallel mean curvature vector field and obtain explicit examples of cmc biconservative surfaces, when n = 3. Also pmc biconservative surfaces with finite total curvature in Hadamard manifolds are considered and a compactness result is obtained, in the last part of the chapter. In the fourth chapter, we study biharmonic submanifolds in Sasakian space forms and obtain classification results, as well as explicit examples, for proper-biharmonic curves, proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in CP n, and 3-dimensional proper-biharmonic integral C-parallel submanifolds in a 7- dimensional Sasakian space form. We also present a method to construct biharmonic anti-invariant submanifolds from biharmonic integral submanifolds. The results in this chapter first appeared in [59], [60], [61], and [65]-[70]. Biharmonic submanifolds in complex space forms are studied in the last chapter. First, are presented some general results on the biharmonicity of certain classes of submanifolds. We continue with a formula that relates the bitension fields of a submanifold in CP n and its corresponding Hopf cylinder in S 2n+1. Next, we prove a result on the biharmonicity of Clifford type submanifolds in CP n, while in the last part of the chapter we classify proper-biharmonic curves and pmc properbiharmonic surfaces in CP n, and also 3-dimensional proper-biharmonic Lagrangian parallel submanifolds in CP 3. This chapter contains results from [64], [70], and [73]. Whilst throughout the thesis we tried to offer the reader an image as complete as possible of our work, due to the need of keeping the presentation at a reasonable length, we were forced to skip many of the proofs and sometimes even not to mention some of our results that otherwise we consider interesting. iii

Rezumat Un subiect, în acelaşi timp clasic şi actual, în geometria diferenţială de astăzi îl reprezintă studiul suprafeţelor de curbură medie constantă (suprafeţe cmc) în spaţii de dimensiune 3, şi deasemeni al cazului mai general al subvarietăţilor având câmpul vectorial curbură medie paralel în fibratul normal (subvarietăţi pmc) în spaţii de dimensiune oarecare. Primele studii de acest fel, consacrate suprafeţelor cmc, au apărut acum mai bine de 60 de ani, în timp ce subvarietăţile pmc au început să câştige interesul lumii matematice la începutul anilor 1970 odată cu apariţia unor articole ca [35] de B.-Y. Chen şi G. D. Ludden, [52, 53] ale lui J. Erbacher, [58] de D. Ferus, [83] de D. A. Hoffman, sau [136] de S.-T. Yau. În continuare, vom trece rapid în revistă doar unele din acele rezultate care au reprezentat (şi încă o fac) o sursă de inspiraţie în activitatea noastră de cercetare. Două instrumente au fost folosite cu precădere în obţinerea acestor rezultate: diferenţialele olomorfe definite pe suprafaţe cmc (sau pmc) precum şi formulele de tip Simons. H. Hopf [85] a folosit pentru prima oară o diferenţială olomorfă pentru a arăta că orice suprafaţă cmc homeomorfă cu o sferă în spaţiul euclidian 3-dimensional este de fapt o sferă euclidiană. Acest rezultat a fost extins de către S.-S. Chern [41] la cazul suprafeţelor cmc în forme spaţiale reale, pentru ca apoi să fie demonstrat pentru suprafeţe cmc în spaţii produs de tipul M 2 (c) R, unde M 2 (c) este o suprafaţă completă şi simplu conexă de curbură constantă c, ca şi pentru cele în Nil(3) şi PSL(2, R), de către U. Abresch şi H. Rosenberg [1, 2]. În mod natural, următorul pas a fost trecerea la studiul suprafeţelor pmc în M n (c) R, unde M n (c) este o formă spaţială reală de curbură secţională constantă c, adică, al acelor suprafeţe care satisfac H = 0, unde este conexiunea din fibratul normal, iar H câmpul vectorial curbură medie. Două din articolele importante dedicate acestui subiect sunt [5, 6] de H. Alencar, M. do Carmo, şi R. Tribuzy. În aceste articole autorii introduc o diferenţială olomorfă (care generalizează diferenţiala Abresch-Rosenberg definită în [1] pentru suprafeţe cmc în M 2 (c) R) şi o folosesc apoi în studiul geometriei unora dintre aceste suprafeţe. Unul dintre rezultatele principale din [6] este o teoremă de reducere a codimensiunii care arată că o suprafaţă pmc în M n (c) R sau este minimală într-o hipersuprafaţă total umbilicală în M n (c); sau este o suprafaţă cmc într-o subvarietate 3-dimensională total umbilicală sau total geodezică în M n (c); sau stă în M 4 (c) R. O altă metodă implicată cu real succes în studiul subvarietăţilor minimale şi, în general, al subvarietăţilor cmc şi pmc, este folosirea ecuaţiilor de tip Simons. În 1968, J. Simons [128] a obţinut expresia laplacianului normei formei a doua fundamentale a unei subvarietăţi minimale într-o varietate riemanniană, pe care apoi a folosit-o în caracterizarea unora dintre aceste subvarietăţi în sfera euclidiană şi în spaţiul euclidian. Un an mai târziu, K. Nomizu şi B. Smyth [112] au generalizat v

vi această formulă în cazul hipersuprafeţelor cmc într-o formă spaţială reală, acest rezultat fiind generalizat la rândul lui, de către B. Smyth s [129], pentru subvarietăţi pmc într-o formă spaţială. De-a lungul anilor astfel de formule, numite astă zi ecuaţii de tip Simons, au fost folosite din ce în ce mai des în articolele dedicate subvarietăţilor cmc şi pmc (vezi, de exemplu, [4, 7, 8, 16, 18, 28, 39, 123]). În ultimii 30 de ani s-a putut observa un interes din ce în ce mai ridicat în studierea unor ecuaţii cu derivate parţiale de ordin 4 care generalizează noţiunea de aplicaţii armonice. Astfel, în influentul lor articol [49], J. Eells şi J. H. Sampson au sugerat noţiunea de aplicaţie biarmonică ψ : M N între două varietăţi riemanniene, definită ca fiind un punct critic al funcţionalei bienergiei E 2 (ψ) = τ(ψ) 2 dv, M unde τ(ψ) = trace ψ dψ este câmpul tensiune al aplicaţiei ψ, ψ fiind conexiunea în fibratul ψ 1 T M. Dacă varietatea M nu este compactă, atunci aplicaţia ψ : M N este, prin definiţie, biarmonică dacă este o soluţie a ecuaţiei Euler-Lagrange τ 2 (ψ) = 0, unde τ 2 (ψ) = τ(ψ) trace R(dψ, τ(ψ))dψ este câmpul bitensiune al lui ψ. Este uşor de văzut că orice aplicaţie armonică este biarmonică, acesta fiind motivul pentru care suntem interesaţi în studierea aplicaţiilor biarmonice care nu sunt armonice, numite aplicaţii propriu-biarmonice. Un caz special este cel al imersiilor biarmonice, sau, altfel spus, al subvarietăţilor biarmonice, adică, al acelor subvarietăţi pentru care aplicaţia de incluziune este biarmonică. În spaţiul euclidian (şi doar în acest spaţiu), această definiţie a subvarietăţilor biarmonice coincide cu cea propusă de către B.-Y. Chen [31], care caracterizează aceste subvarietăţi prin faptul că au câmpul vectorial curbură medie armonic. Deşi în spaţiul euclidian au fost obţinute doar rezultate care sugerează că nu există exemple de subvarietăţi propriu-biarmonice (vezi, de exemplu, [34, 48, 91, 92, 106]), în spaţii de curbură secţională diferită de zero, au fost găsite numeroase exemple şi demonstrate multe rezultate de clasificare a unor astfel de subvarietăţi, în articole cum ar fi [12]-[14], [21]-[26], [44, 88, 100, 101, 104, 106], [117]-[119], [124]-[126] şi [139]. Această teză este structurată în două părţi, prima, constând din două capitole, fiind dedicată subvarietăţilor pmc, în timp ce a doua, formată din trei capitole, are drept scop prezentarea rezultatelor obţinute în studiul subvarietăţilor biarmonice. În primul capitol, prezentăm rezultate din [62], [77] şi [78] privind suprafeţele pmc în forme spaţiale complexe, cosimplectice şi respectiv sasakiene. În toate aceste situaţii demonstrăm teoreme de reducere a codimensiunii şi introducem diferenţiale olomorfe pe care le folosim în studierea geometriei unora dintre aceste suprafeţe. Al doilea capitol este dedicat studiului subvarietăţilor pmc de dimensiune arbitrară în M n (c) R. Mai întâi demonstrăm două ecuaţii de tip Simons pe care apoi le folosim pentru a găsi rezultate de tip gap pentru astfel de subvarietăţi. Ne îndreptăm atenţia şi spre suprafeţele pmc având curbura totală finită şi demonstrăm o teoremă despre la compactitatea acestora. Capitolul se încheie cu un rezultat de clasificare a suprafeţelor pmc care fac un unghi constant cu ξ, câmpul vectorial unitar tangent la dreapta reală. Aceste rezultate au fost obţinute în [17], [63], [72], [74], [75] şi [76].

Începem prezentarea subvarietăţilor biarmonice în capitolul al treilea, construit pe articolele [71] şi [72]. Prezentăm mai întâi o teoremă de tip gap pentru subvarietăţi pmc propriu-biarmonice în M n (c) R şi deasemeni clasificăm suprafeţele pmc propriu-biarmonice în acest spaţiu. Un alt subiect al acestui capitol îl reprezintă suprafeţele biconservative în M n (c) R, adică, acele suprafeţe pentru care partea tangentă a câmpului bitensiune se anulează (menţionăm că subvarietăţile biconservative au început să fie studiate foarte recent în articole cum ar fi [27, 80, 107, 108]). Aici, determinăm ecuaţia explicită a suprafeţelor pmc biconservative şi găsim exemple explicite de suprafeţe cmc biconservative în cazul când n = 3. Încheiem capitolul cu un rezultat privind compactitatea suprafeţelor pmc biconservative de curbură totală finită în varietăţi Hadamard. În al patrulea capitol, studiem subvarietăţile biarmonice în forme spaţiale sasakiene şi obţinem atât rezultate de clasificare, cât şi exemple explicite, pentru curbe propriu-biarmonice, cilindri Hopf propriu-biarmonici peste hipersuprafeţe omogene reale în CP n şi pentru subvarietăţi propriu-biarmonice integrale C-paralele de dimensiune 3 într-o formă spaţială sasakiană 7-dimensională. Deasemeni, prezentăm o metodă prin care pot fi obţinute subvarietăţi biarmonice anti-invariante pornind de la subvarietăţi biarmonice integrale. Rezultatele cuprinse în acest capitol au apărut în [59], [60], [61] şi [65]-[70]. Subvarietăţile biarmonice în forme spaţiale complexe sunt studiate în ultimul capitol al tezei. Mai întâi prezentăm unele rezultate generale cu privire la biarmonicitatea unor clase speciale de subvarietăţi în aceste spaţii. Continuăm cu o formulă de legătură între câmpurile bitensiune ale unei subvarietăţi în CP n şi cilindrului Hopf corespunzător în S 2n+1. În continuare, caracterizăm subvarietăţile de tip Clifford propriu-biarmonice în CP n, iar în ultima parte a capitolului clasificăm curbele propriu-biarmonice şi suprafeţele pmc propriu-biarmonice în CP n şi deasemeni subvarietăţile propriu-biarmonice lagrangiene paralele 3-dimensionale în CP 3. Acest capitol conţine rezultate obţinute în [64], [70] şi [73]. vii

Contents Abstract Rezumat i v Part 1. Submanifolds with Parallel Mean Curvature 1 Chapter 1. Reduction of Codimension Results and Holomorphic Differentials for Surfaces with Parallel Mean Curvature 3 1. Introduction 3 2. Surfaces with parallel mean curvature in complex space forms 3 3. Surfaces with parallel mean curvature in CP n R and CH n R 12 4. Surfaces with parallel mean curvature in Sasakian space forms 24 Chapter 2. Simons Type Formulas and Applications. Surfaces with Parallel Mean Curvature and Finite Total Curvature 39 1. Introduction 39 2. Preliminaries 39 3. Simons type formulas 41 4. Gap theorems for submanifolds with parallel mean curvature in M n (c) R 50 5. Holomorphic differentials and Simons type formulas. Surfaces with constant mean curvature and finite total curvature 59 6. Helix surfaces with parallel mean curvature in M n (c) R 66 Part 2. Biharmonic Submanifolds 71 Chapter 3. Biharmonic and Biconservative Submanifolds in M n (c) R 73 1. Introduction 73 2. Preliminaries 73 3. A gap theorem for biharmonic submanifolds with parallel mean curvature in S n R 75 4. Biharmonic pmc surfaces in S n (c) R 79 5. Biconservative surfaces with parallel mean curvature in M n (c) R 81 6. Biconservative surfaces with constant mean curvature in M 3 (c) R 86 7. Biconservative surfaces with constant mean curvature in Hadamard manifolds 90 Chapter 4. Biharmonic Submanifolds in Sasakian Space Forms 95 1. Introduction 95 2. Biharmonic Legendre curves in Sasakian space forms 95 3. Biharmonic non-legendre curves in Sasakian space forms 102 4. A non-existence result for biharmonic curves in S 7 109 ix

x 5. Biharmonic anti-invariant submanifolds in Sasakian space forms 111 6. Biharmonic hypersurfaces in Sasakian space forms 116 7. Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms 120 Chapter 5. Biharmonic Submanifolds in Complex Space Forms 129 1. Introduction 129 2. General biharmonicity results for submanifolds in complex space forms 129 3. The Hopf fibration and the biharmonic equation 132 4. Biharmonic submanifolds of Clifford type in CP n 135 5. Biharmonic curves in CP n 138 6. Biharmonic surfaces with parallel mean curvature in complex space forms 144 7. Biharmonic parallel Lagrangian submanifolds of CP 3 153 Further Developments 157 Index 159 Bibliography 161

Part 1 Submanifolds with Parallel Mean Curvature

CHAPTER 1 Reduction of Codimension Results and Holomorphic Differentials for Surfaces with Parallel Mean Curvature 1. Introduction We consider surfaces with parallel mean curvature vector field (pmc surfaces) in N, where N is either a complex space form, or a cosymplectic space form, or a Sasakian space form. These three cases are treated in three separate sections. In each of these sections, we prove reduction of codimension results (Theorems 1.15, 1.35, and 1.56) and study the geometry of some of these surfaces. The main tool employed in these studies are some holomorphic differentials introduced by Theorems 1.4, 1.19, and 1.57. Using these holomorphic differentials, we classify pmc 2-spheres in 2-dimensional complex space forms (Theorem 1.5) and anti-invariant pmc 2-spheres in cosymplectic space forms (Theorems 1.40, 1.42, and 1.45), prove non-existence results for pmc 2-spheres with constant Kähler angle in complex space forms (Theorem 1.17) and anti-invariant pmc 2-spheres in Sasakian space forms (Theorem 1.68), and determine all integral complete pmc surfaces that are pseudoumbilical and have non-negative Gaussian curvature in 7-dimensional Sasakian space forms (Theorem 1.63). 2. Surfaces with parallel mean curvature in complex space forms 2.1. Preliminaries. Let Σ m be an isometrically immersed submanifold in a Riemannian manifold N. The second fundamental form σ of Σ m is then defined by the equation of Gauss (1.1) N XY = X Y + σ(x, Y ), while the shape operator A and the normal connection are given by the equation of Weingarten (1.2) N XV = A V X + XV, for any vector fields X and Y tangent to Σ m and any normal vector field V, where N and are the Levi-Civita connections on N and Σ m, respectively. The mean curvature vector field H of Σ m is given by H = (1/m) trace σ. We also have the Gauss equation of Σ m in N (1.3) R(X, Y )Z, W = R N (X, Y )Z, W + σ(y, Z), σ(x, W ) σ(x, Z), σ(y, W ), the Codazzi equation (1.4) (R N (X, Y )Z) = ( Xσ)(Y, Z) ( Y σ)(x, Z), 3

4 Reduction of Codimension and Holomorphic Differentials and the equation of Ricci (1.5) R (X, Y )U, V = [A U, A V ]X, Y + R N (X, Y )U, V, for any vector fields X, Y, Z, and W tangent to Σ m and any normal vector fields U and V, where R N, R, and R are the curvature tensors corresponding to N,, and, respectively. Definition 1.1. If the mean curvature vector field H of Σ m is parallel in the normal bundle, i.e., H = 0, then Σ m is called a pmc submanifold. Definition 1.2. If the mean curvature H of Σ m is constant, then Σ m is called a cmc submanifold. Definition 1.3. If the second fundamental form σ of Σ m satisfies σ = 0, then Σ m is called a parallel submanifold. Now, let N n (c) be a complex space form with complex dimension n, complex structure (J,, ), and constant holomorphic sectional curvature c. Then N n (c) either is CP n (c), or C n, or CH n (c), as c > 0, c = 0, or c < 0, respectively (see [135]). The curvature tensor of N n (c) is given by (1.6) R N (U, V )W = c { V, W U U, W U + JV, W JU JU, W JV 4 + 2 JV, U JW }. 2.2. A holomorphic differential. We will consider pmc surfaces Σ 2 in the complex space form N n (c) and introduce a holomorphic differential defined on such surfaces, that will be used to characterize the pmc 2-spheres in the 2-dimensional complex space forms and also to prove the non-existence of non-pseudo-umbilical pmc 2-spheres with constant Kähler angle in N n (c). Theorem 1.4 ([62]). Let Σ 2 be a pmc surface in a complex space form N n (c). Then the (2, 0)-part of the quadratic form Q defined on Σ 2 by is holomorphic. Q(X, Y ) = 8 H 2 σ(x, Y ), H + 3c JX, H JY, H, Proof. First, let us consider isothermal coordinates (u, v) on Σ 2. Then we have ds 2 = λ 2 (du 2 + dv 2 ) and define z = u + iv, z = u iv, dz = 1 (du + idv), d z = 1 (du idv) 2 2 and Z = 1 2 ( ), Z = 1 2 ( u i v u + i v It is easy to see that Z, Z = / u, / u = / v, / v = λ 2. In the following, we will show that Z(Q(Z, Z)) = 0. First, since ZZ = 0, we have ). ( Zσ)(Z, Z) = Zσ(Z, Z) 2σ( ZZ, Z) = Zσ(Z, Z), and then, since H is parallel, Z( σ(z, Z), H ) = N Z σ(z, Z), H + σ(z, Z), N Z H = ( Zσ)(Z, Z), H.

(1.7) Reduction of Codimension and Holomorphic Differentials 5 Now, from the Codazzi equation (1.4), again using H = 0, we obtain Z( σ(z, Z), H ) = ( Zσ)( Z, Z), H + (R N ( Z, Z)Z), H Next, since Z Z = 0, we have + σ(z, Z), ZH = ( Zσ)( Z, Z), H + R N ( Z, Z)Z, H. ( Zσ)( Z, Z) = Zσ( Z, Z) σ( Z, Z Z) and, therefore, using σ( Z, Z) = Z, Z H, Z Z = (1/λ 2 ) Z Z, Z Z, and H = 0, we get (1.8) ( Zσ)( Z, Z), H = 0. We can also easily check that (JZ) = (1/λ 2 ) JZ, Z Z and then, since N J = 0, N Z Z = σ( Z, Z) = Z, Z H, and H = 0, we have (1.9) Z( JZ, H 2 ) = 2 H 2 Z, JZ JZ, H Finally, from the expression (1.6) of the curvature tensor field R N, we see that (1.10) R N ( Z, Z)Z, H = 3c 4 Z, JZ H, JZ. We conclude replacing (1.8), (1.9), and (1.10) into (1.7). 2.3. Pmc surfaces in 2-dimensional complex space forms. In the following, for pmc surfaces Σ 2 in N 2 (c), with H 0, we will introduce another quadratic form Q with holomorphic (2, 0)-part. Then, using Q and Q, we will classify the non-minimal pmc 2-spheres in N 2 (c). Let us consider a pmc surface Σ 2 in N 2 (c), a local orthonormal frame {Ẽ1, Ẽ2} on Σ 2, and denote by θ the Kähler angle function defined by JẼ1, Ẽ2 = cos θ. The immersion x : Σ 2 N is said to be holomorphic if cos θ = 1, anti-holomorphic if cos θ = 1, and totally real if cos θ = 0. In the following we shall assume that x is neither holomorphic or anti-holomorphic. Next, we take E 3 = (H/ H ) and let E 4 be the unique unit normal vector field orthogonal to E 3 compatible with the orientation of Σ 2 in N. Since E 3 is parallel in the normal bundle so is E 4, and, as the Kähler angle is independent of the choice of the orthonormal frame on the surface (see [42]), we have JE 4, E 3 = cos θ. Now, we consider the vector fields E 1 = cot θe 3 1 sin θ JE 4 and E 2 = 1 sin θ JE 3 + cot θe 4 tangent to the surface and get an orthonormal frame field {E 1, E 2, E 3, E 4 } along Σ 2 in N. We define a new quadratic form Q on Σ 2 by Q (X, Y ) = 8i H σ(x, Y ), E 4 + 3c JX, E 4 JY, E 4. In the same way as in the case of Q, also using Z, JZ = i Z, Z E 1, JE 2 = i Z, Z cos θ, it follows that also the (2, 0)-part of Q is holomorphic.

6 Reduction of Codimension and Holomorphic Differentials In order to classify pmc 2-spheres in N 2 (c), we will use a result of T. Ogata in [113], that we will briefly recall in the following (see also [81, 94]). Using the local orthonormal frame field {E 1, E 2, E 3, E 4 } and considering isothermal coordinates (u, v) on the surface, T. Ogata proved that there exist complex-valued functions a and d on Σ 2 such that θ, λ, a, and d satisfy θ λ z = λ(a + b), z = λ 2 (ā b) cot θ a (1.11) z = λ ( ) 2 a 2 2ab + 3c sin2 θ 8 cot θ d z = 2λ(a b)d cot θ, d 2 = a 2 + c(3 sin2 θ 2) 8 where z = u + iv and H = 2b; and also the converse: if c is a real constant, b a positive constant, Σ 2 a 2-dimensional Riemannian manifold, and there exist some functions θ, a, and d on Σ 2 satisfying (1.11), then there is an isometric immersion of Σ 2 into N 2 (c) with with Kähler angle θ and parallel mean curvature vector field of length equal to 2b. The second fundamental form of Σ 2 in N with respect to {E 1, E 2, E 3, E 4 } is given by 2b R(ā + d) I(ā + d) I(ā d) R(ā d) σ 3 =, σ 4 =, I(ā + d) 2b + R(ā + d) R(ā d) I(ā d) where R and I denote the real and the imaginary parts, respectively, of a complex number, and the Gaussian curvature of Σ 2 is K = 4b 2 4 d 2 + c/2 (see also [81]). Assume now that the (2, 0)-part of Q and that of Q vanish on the surface Σ 2. It follows, from the expression of the second fundamental form, that d + a R, d a R, and 32b( d + a) 3c sin 2 θ = 0, 32b( d a) + 3c sin 2 θ = 0. Therefore d = 0 and a = 3c sin 2 θ/(32b) and, from the fifth equation of (1.11), we get (1.12) 9c 2 sin 4 θ + 128cb 2 (3 sin 2 θ 2) = 0. We have to split the study of this equation in two cases. First, if c = 0, then (1.12) holds and also a = 0. Next, if c 0, it follows that θ is a constant function. This, together with the first equation of (1.11), leads to a = 3c sin 2 θ/(32b) = b. Replacing in equation (1.12), we obtain c = 12b 2 and then sin 2 θ = 8/9. We note that in both cases the Gaussian curvature of Σ 2 is given by K = 4b 2 +c/2 = constant (see [81]). Then, using [81, Theorem 1.1], we obtain the following classification result. Theorem 1.5 ([62]). If the (2, 0)-part of Q and the (2, 0)-part of Q vanish on a pmc surface Σ 2 in N 2 (c), with H = 2b > 0, then either (1) N 2 (c) = CH 2 ( 12b 2 ) and Σ 2 is the slant surface in [32, Theorem 3(2)]; (2) N 2 (c) = C 2 and Σ 2 is a part of a round sphere in a hyperplane in C 2. Since the Gaussian curvature K is nonnegative only in the second case of Theorem 1.5, we have also recovered a result in [81]. Corollary 1.6 ([81]). If Σ 2 is a non-minimal pmc 2-sphere in a 2-dimensional complex space form, then it is a round sphere in a hyperplane in C 2.

Reduction of Codimension and Holomorphic Differentials 7 2.4. Reduction of codimension. Let Σ 2 be a surface in a complex space form N n (c), n 3 and c 0, with parallel mean curvature vector field H 0. We will prove that either Σ 2 is pseudo-umbilical or it lies in a complex space form with the same holomorphic sectional curvature c and complex dimension 5. First, since H is parallel, using the Ricci equation (1.5) of Σ 2 in N and the expression (1.6) of the curvature of the complex space form, we immediately have the following lemma. Lemma 1.7. For any vector V normal to Σ 2, which is also orthogonal to JT Σ 2 and to JH, we have [A H, A V ] = 0, i.e., A H commutes with A V. Corollary 1.8 ([62]). At any point p Σ 2 either H is an umbilical direction, or there exists a basis that diagonalizes simultaneously A H and A V, for all normal vectors satisfying V JH, if n = 3 and H JT Σ 2, or the conditions in Lemma 1.7, otherwise. Proposition 1.9 ([62]). Assume that H is nowhere an umbilical direction. Then there exists a parallel subbundle of the normal bundle that contains the image of the second fundamental form σ and has dimension less or equal to 8. Proof. We consider a subbundle L of the normal bundle, given by L = span{im σ (J Im σ) (JT Σ 2 ) }, where (J(Im σ)) = {(Jσ(X, Y )) : X, Y tangent to Σ 2 }, (J(T Σ 2 )) = {(JX) : X tangent to Σ 2 }, and we will show that L is parallel. First, we will prove that, if V is orthogonal to L, then E i V is orthogonal to JT Σ 2 and to JH, where {E 1, E 2 } is a local orthonormal tangent frame field with respect to which we have σ(e 1, E 2 ), V = σ(e 1, E 2 ), H = 0. We get, indeed, and (JH), E i V = (JH), N E i V = N E i (JH), V = N E i JH, V + N E i (JH), V = JA H E i, V + σ(e i, (JH) ), V =0 (JE j ), E i V = N E i (JE j ), V = N E i JE j, V + N E i (JE j ), V = J Ei E j, V Jσ(E i, E j ), V + σ(e i, (JE j ) ), V =0. Next, we shall prove that if a normal subbundle S is orthogonal to L, then so is S, i.e., σ(e i, E j ), E k V = 0, Jσ(E i, E j ), E k V = 0, and JE i, E k V = 0 for any V S and i, j, k {1, 2}. We only have to verify the first two of these properties. Denote by A ijk = E k σ(e i, E j ), V and, since σ is symmetric, notice that A ijk = A jik. We also have A ijk = σ(e i, E j ), E k V, since V is orthogonal to L.

8 Reduction of Codimension and Holomorphic Differentials Now, we have ( E k σ)(e i, E j ), V = E k σ(e i, E j ), V σ( Ek E i, E j ), V σ(e i, Ek E j ), V = E k σ(e i, E j ), V, and, using the Codazzi equation (1.4), ( E k σ)(e i, E j ), V = ( E i σ)(e k, E j ) + (R N (E k, E i )E j ), V = ( E j σ)(e k, E i ) + (R N (E k, E j )E i ), V = ( E i σ)(e k, E j ), V = ( E j σ)(e k, E i ), V, that shows that A ijk = A kji = A ikj. Next, since E k V is orthogonal to JT Σ 2 and to JH, it follows that {E 1, E 2 } diagonalizes A Ek V and we get A ijk = σ(e i, E j ), E k V = E i, A Ek V E j = 0 for any i j. Hence, A ijk = 0 if two indices are different from each other. Finally, we have A iii = σ(e i, E i ), E i V = 2H, E i V + σ(e j, E j ), E i V = 2 E i H, V A jji = 0. It is easy to see that, if V is orthogonal to L, then JV is normal and orthogonal to L. It follows that (Jσ(E i, E j )), E k V = N E k (Jσ(E i, E j )), V and we conclude. = N E k Jσ(E i, E j ), V + N E k (Jσ(E i, E j )), V = JA σ(ei,e j )E k, V J E k σ(e i, E j ), V + σ(e k, (Jσ(E i, E j )) ), V = E k σ(e i, E j ), JV = 0 From Proposition 1.9, it is easy to see that, when H is not umbilical, T Σ 2 L is parallel with respect to the Levi-Civita connection on N. Since we also have J(T Σ 2 L) T Σ 2 L along the surface, and then R N (X, Y )Z T Σ 2 L for any X, Y, Z T Σ 2 L, we can apply [54, Theorem 2] to conclude that there exists a totally geodesic submanifold N 5 (c) of N n (c) that contains our surface. Proposition 1.10 ([62]). If Σ 2 is not pseudo-umbilical, then it lies in a complex space form N 5 (c). When H is umbilical we use the quadratic form Q to prove the following proposition. Proposition 1.11 ([62]). Let Σ 2 be a pmc surface in N n (c), with c 0 and H 0. If H is umbilical, then Σ 2 is totally real. Proof. Since H is umbilical, it follows that σ(z, Z), H = 0, which implies Q(Z, Z) = 3c JZ, H 2.

Reduction of Codimension and Holomorphic Differentials 9 Next, since Z(Q(Z, Z)) = 0, we have 0 = Z( JZ, H 2 ) = 2 H 2 JZ, H JZ, Z. Hence, JZ, Z = 0 or JZ, H = 0. Assume that the set of zeroes of equation JZ, Z = 0 is not Σ 2. Then, it is a closed set without interior points and its complement is an open dense set in Σ 2. In this last set we have JZ, H = 0 and then, since H is parallel and Σ 2 is pseudo-umbilical, 0 = Z( JZ, H ) = J N Z Z, H + JZ, N Z H which means that Σ 2 is totally real. = Z, Z JH, H JZ, A H Z = H 2 JZ, Z, Remark 1.12. Some kind of a converse result was obtained by B. Y. Chen and K. Ogiue [36]. They proved that, if a unit normal vector field to a totally real 2-sphere in a complex space form is parallel and isoperimetric, then it is umbilical. Remark 1.13. N. Sato [127] proved that, if Σ m is a pseudo-umbilical submanifold of a complex space form, with nonzero parallel mean curvature vector field, then it is a totally real submanifold. Moreover, the mean curvature vector field H is orthogonal to JT Σ m. Therefore, the (2, 0)-part of Q, defined on Σ 2, vanishes. Remark 1.14. Since the map p Σ 2 (A H µ I)(p), where µ is a constant, is analytic, it follows that if H is an umbilical direction, then this either holds on Σ 2, or only for a closed set without interior points. In this second case H is not an umbilical direction in an open dense set. Consequently, only the two above studied cases can occur. From Propositions 1.10 and 1.11, it follows the main result of this section. Theorem 1.15 ([62]). Let Σ 2 be a non-minimal pmc surface in a complex space form N n (c), n 3, c 0. Then, one of the following holds: (1) Σ 2 is a totally real pseudo-umbilical surface; or (2) Σ 2 is not pseudo-umbilical and lies in a complex space form N 5 (c). Remark 1.16. The case when c = 0 was treated in [136, Theorem 4]. 2.5. A non-existence result for pmc 2-spheres with constant Kähler angle. We consider pmc surfaces Σ 2 in a complex space form N n (c), n 3, c 0, with constant Kähler angle and H 0, on which the (2, 0)-part of Q vanishes. We will compute the Laplacian of A H 2 and show that there are no 2-spheres with all these properties. Let {E 1, E 2 } be a local orthonormal frame field on Σ 2 such that H JE 1. The fact that the (2, 0)-part of the quadratic form Q vanishes can be written as { 8 H 2 σ(e 1, E 1 ) σ(e 2, E 2 ), H = 3c( JE 1, H 2 JE 2, H 2 ) (1.13) 8 H 2 σ(e 1, E 2 ), H = 3c JE 1, H JE 2, H, and, from the second equation, we see that σ(e 1, E 2 ), H = 0. It follows that {E 1, E 2 } diagonalizes simultaneously A H and A V, for all normal vectors V as in Corollary 1.8, since we are in the second case of Theorem 1.15 (see Remark 1.12).

10 Reduction of Codimension and Holomorphic Differentials Next, since Σ 2 is not holomorphic or anti-holomorphic, we have that cos θ ±1 on an open dense set, where θ is the Kähler angle. We again consider the normal vector fields (1.14) E 3 = cot θe 1 1 sin θ JE 2 and E 4 = 1 sin θ JE 1 cot θe 2, with the property that {E 1, E 2, E 3, E 4 } is a basis in span{e 1, E 2, JE 1, JE 2 }. It is easy to see that, if H JT Σ 2, then our surface is pseudo-umbilical, which is a contradiction. On the other hand, if we assume that H span{e 3, E 4 } it follows H = ± H E 3, since JE 1 H, and then E 3 is parallel. Also, since all normal vectors but E 4 verify conditions in Corollary 1.8 we have σ(e 1, E 2 ) E 4. By using these facts and the expression of E 3, we obtain σ(e i, E j ) span{e 3, E 4 } for i, j {1, 2}, and then dim L = 2, where L is the subbundle in Proposition 1.10. Therefore, again using [54, Theorem 2], we get that Σ 2 lies in a complex space form N 2 (c), a case that we have already studied. In the following, we will assume that H / span{e 3, E 4 }, and, since we also know that H is not orthogonal to JT Σ 2, one obtains that H can be written as H = H (cos βe 3 + sin βe 5 ), where β is a real-valued function defined locally on Σ 2 and E 5 is a unit normal vector field such that E 5 JT Σ 2. We consider the local orthonormal frame field {E 1, E 2, E 3, E 4, E 5, E 6 = JE 5,..., E 2n 1, E 2n = JE 2n 1 } on N along Σ 2 and its dual frame {ω i } 2n i=1. The vector fields E i are well defined at the points of Σ 2 where sin(2β) 0, which, due to our assumptions, form an open dense set in Σ 2. The structure equations of the surface are dφ = iω 12 φ and dω 12 = i Kφ φ, 2 where φ = ω 1 + iω 2, the real 1-form ω 12 is the connection form of the Riemannian metric on Σ 2, and K is the Gaussian curvature of the surface. A result of T. Ogata [114], together with H E i, i 4, i 5, implies that, with respect to the above local orthonormal frame field, the components of the second fundamental form are given by H cos β R(ā + c) I(ā + c) σ 3 = I(ā + c) H cos β + R(ā + c) I(ā d) R(ā d) σ 4 = R(ā d) I(ā d) σ 5 = H sin β R(ā 3 + d 3 ) I(ā 3 + d 3 ) I(ā 3 + d 3 ) H sin β + R(ā 3 + d 3 ) σ 2α 1 = R(ā α + d α ) I(ā α + d α ) I(ā α + d α ) R(ā α + d α )

Reduction of Codimension and Holomorphic Differentials 11 σ 2α = I(ā α d α ) R(ā α d α ) R(ā α d α ) I(ā α d α ) where a, d, a α, d α, with α {3,..., n}, are complex-valued functions defined locally on Σ 2. We note that, since σ(e 1, E 2 ) H and σ(e 1, E 2 ) E 5, it follows σ(e 1, E 2 ) E 3. Moreover, since σ(e 1, E 2 ) E i for any i {1,..., 2n} \ {4, 6}, we have ā + d R, ā 3 + d 3 R, and a α = d α for any α 4. In the same paper [114], the author computed the differential of the Kähler angle function θ for a minimal surface. In the same way, this time for our surface, we get ( 0 = dθ = a H ) ( 2 cos β φ + ā H ) 2 cos β φ. The next step is to determine the connection form ω 12 and the differential of the function β by using that H is parallel. We have (1.15) E i H = ( sin βe 3 + cos βe 5 )E i (β) + cos β E i E 3 + sin β E i E 5 = 0 for i {1, 2}, and then cos β N E i E 3, E 4 + sin β N E i E 3, E 4 = 0, from where, using equation (1.14) and the expression of the second fundamental form, we get ω 12 (E 1 ) = cot θi(ā d) tan β sin θ I(ā 3 d 3 ) ω 12 (E 2 ) = H cot θ ( ( θ ( θ ) cos β 2 cot θra + tan β tan Ra 3 cot 3 2) 2)Rd and, therefore, ω 12 = f 1 φ + f 1 φ, where (1.16) f 1 = i ( H cot θ tan β + 2 cot θa 2 cos β sin θ (a 3 d 3 ) + cot θ tan β(a 3 + d ) 3 ). Now, from equation (1.15), we also obtain E i (β) + N E i E 3, E 5 = 0, i {1, 2} and then, using (1.14) and the expression of the second fundamental form, one obtains ( θ ( θ E 1 (β) = H cot θ sin β + tan Ra 3 cot Rd 3 2) 2) and dβ(e 2 ) = 1 sin θ I(ā 3 d 3 ). Hence, the differential of β is given by dβ = f 2 φ + f 2 φ, where (1.17) f 2 = 1 ( H cot θ sin β + 1 2 sin θ (a 3 d 3 ) cot θ(a 3 + d ) 3 ). We note that, since the Kähler angle θ is constant, we have a = ā = ( H /2) cos β, and then, from (1.16), it follows (1.18) f 1 = i { ( H cot θ 2 cos β + 1 cos β ) tan β sin θ (a 3 d 3 ) + cot θ tan β(a 3 + d } 3 ). Let us now return to the first equation of (1.13), which can be rewritten as µ 1 µ 2 = 3 8 c sin2 θ cos 2 β,

12 Reduction of Codimension and Holomorphic Differentials where A H E i = µ i E i. Since µ 1 + µ 2 = 2 H 2, we have Thus, we have µ 1 = H 2 + 3 16 c sin2 θ cos 2 β and µ 2 = H 2 3 16 c sin2 θ cos 2 β. (1.19) A H 2 = µ 2 1 + µ 2 2 = 2 H 4 + 9 128 c2 sin 4 θ cos 4 β, and then A H 2 = 9 128 c2 sin 4 θ (cos 4 β). After a straightforward computation, using (1.17) and (1.18), we get ( (cos 4 β) = 4 cos 4 β K + 4 f 1 2 + 12 if 1 + H cot θ 2 ), cos β which means that A H 2 = 9 ( 32 c2 sin 4 θ cos 4 β K + 4 f 1 2 + 12 if 1 + H cot θ 2 ). cos β Assume now that Σ 2 is complete and has nonnegative Gaussian curvature. It follows, from a result of A. Huber in [87], that Σ 2 is parabolic. From the above formula, we get that A H 2 is a subharmonic function, and, since A H 2 is bounded by (1.19), it follows that K = 0, which leads to the following non-existence result. Theorem 1.17 ([62]). There are no non-minimal pmc 2-spheres with constant Kähler angle in a non-flat complex space form. 3. Surfaces with parallel mean curvature in CP n R and CH n R 3.1. Preliminaries. Let M n (c) be a complex space form with the complex structure (J,, M ), consider the product manifold N 2n+1 = M n (c) R and define the following tensors on N 2n+1 : ϕ = J dπ, ξ = t, η = dt, and, N =, M + dt dt, where π : M n (c) R M n (c) is the projection map and t is the standard coordinate function on the real axis. Then (N 2n+1, ϕ, ξ, η,, N ) is a cosymplectic space form with constant ϕ-sectional curvature equal to c (see [3, 20]). We shall explain what this means in the following. An almost contact metric structure on an odd-dimensional manifold N 2n+1 is given by (ϕ, ξ, η,, ), where ϕ is a tensor field of type (1, 1) on N, ξ is a vector field, η is its dual 1-form, and, is a Riemannian metric such that ϕ 2 U = U + U, ξ ξ and ϕu, ϕv = U, V η(u)η(v ), for all tangent vector fields U and V. An almost contact metric structure (ϕ, ξ, η,, ) is called normal if N ϕ (U, V ) + 2dη(U, V )ξ = 0, where N ϕ (U, V ) = [ϕu, ϕv ] ϕ[ϕu, V ] ϕ[u, ϕv ] + ϕ 2 [U, V ] is the Nijenhuis tensor field of ϕ.

Reduction of Codimension and Holomorphic Differentials 13 Definition 1.18. An almost contact metric manifold (N, ϕ, ξ, η,, ) is a cosymplectic manifold if it is normal and both the 1-form η and the fundamental 2-form Ω, defined by Ω(U, V ) = U, ϕv, are closed. Equivalently, an almost contact metric manifold is cosymplectic if and only if ϕ is parallel, i.e., N ϕ = 0, where N is the Levi-Civita connection. This implies that the vector field ξ and the 1-form η are also parallel. We note that a cosymplectic manifold has a natural local product structure as a product of a Kähler manifold and a 1-dimensional manifold, but there exist compact cosymplectic manifolds which are not global products (see [19, 43]). We also recall that a submanifold Σ m of an almost contact metric manifold is called invariant when ϕ(t Σ m ) T Σ m and antiinvariant when ϕ(t Σ m ) NΣ m, where NΣ m is the normal bundle of Σ m. Let (N, ϕ, ξ, η,, ) be an almost contact metric manifold. The sectional curvature of a 2-plane generated by U and ϕu, where U is a unit vector orthogonal to ξ, is called ϕ-sectional curvature determined by U. A cosymplectic manifold with constant ϕ-sectional curvature c is called a cosymplectic space form and is denoted by N(c). The curvature tensor field of a cosymplectic space form N(c) is given by (1.20) R N (U, V )W = c { V, W U U, W V + U, ϕw ϕv V, ϕw ϕu 4 + 2 U, ϕv ϕw + η(u)η(w )V η(v )η(w )U + U, W η(v )ξ V, W η(u)ξ}. 3.2. A holomorphic differential. Although our main interest is to study pmc surfaces in product spaces of type M n (c) R, where M n (c) is a complex space form, it is more convenient to treat the more general case where the surfaces are immersed in an arbitrary cosymplectic space form. Let us consider N 2n+1 (c) a cosymplectic space form endowed with the cosymplectic structure (ϕ, ξ, η,, ) and constant ϕ-sectional curvature c. Working in the same way as in the proof of Theorem 1.4, one obtains the following result. Theorem 1.19 ([77]). If Σ 2 is a pmc surface in a cosymplectic space form N 2n+1 (c), then the (2, 0)-part of the quadratic form Q, defined on Σ 2 by Q(X, Y ) = 8 H 2 σ(x, Y ), H c H 2 η(x)η(y ) + 3c ϕx, H ϕy, H, is holomorphic. 3.3. Some examples of pmc surfaces. Let us first recall the definition of Frenet curves in a Riemannian manifold. Definition 1.20. Let γ : I R M n be a curve parametrized by arc-length in a Riemannian manifold. The curve γ is called a Frenet curve of osculating order r, 1 r n, if there exist r orthonormal vector fields {E 1 = γ,..., E r }, along γ, such that M E 1 E 1 = κ 1 E 2, M E 1 E i = κ i 1 E i 1 + κ i E i+1, M E 1 E r = κ r 1 E r 1, for i {2,..., r 1}, where {κ 1,..., κ r 1 } are positive functions on I called the curvatures of γ. A Frenet curve of osculating order r is called a helix of order r if κ i = constant > 0, 1 i r 1. A helix of order 2 is called a circle, and a helix of order 3 is simply called a helix.

14 Reduction of Codimension and Holomorphic Differentials When γ is a Frenet curve in a complex space form M n (c), then its complex torsions are defined by τ ij = E i, JE j, 1 i < j r, where (J,, ) is the complex structure on M n (c). A helix of order r is called a holomorphic helix of order r if all complex torsions are constant. It is easy to see that a circle is always a holomorphic circle (see [103]). Definition 1.21. Let M be a Riemannian manifold and consider the product manifold M R. A submanifold Σ m in M R is called a vertical cylinder over Σ m 1 if Σ m = π 1 (Σ m 1 ), where π : M R M is the projection map and Σ m 1 is a submanifold in M. It is easy to see that vertical cylinders Σ m = π 1 (Σ m 1 ) are characterized by the fact that the unit vector field ξ tangent to R is also tangent to Σ m. In order to find examples of pmc surfaces we will focus our attention on vertical cylinders Σ 2 = π 1 (γ) in product spaces M n (c) R, where π : M n (c) R M n (c) is the projection map and γ : I M n (c) is a Frenet curve of osculating order r in M n (c). For any vector field X tangent to M n (c) we shall denote by X H its horizontal lift to M n (c) R. As for the Riemannian metrics on M n (c) and M n (c) R, we will use the same notation,. Obviously, {E H 1, ξ} is a local orthonormal frame on Σ2 and E H i, 1 < i r, are normal vector fields. Then the mean curvature vector H is given by H = 1 2 (σ(eh 1, E H 1 ) + σ(ξ, ξ)) = 1 2 κ 1E H 2, where κ 1 = κ 1 π and we used the first Frenet equation for γ and O Neill s equation ([110]) in the case of cosymplectic space forms, i.e., N X H Y H = ( M X Y )H, for any vector fields X and Y tangent to M n (c) (see also [3]). Next, from the second Frenet equation, we have (1.21) N E H 1 H = 1 2 ( M E 1 (κ 1 E 2 )) H = 1 2 (κ 1E 2 κ 2 1E 1 + κ 1 κ 2 E 3 ) H. It is easy to verify that ξ E1 H = E1 H ξ = 0, where is the connection on the surface, and then we get that [ξ, E1 H] = 0, which means N ξ EH 1 = N ξ = 0. Now, E1 H since from (1.20) it follows that R N (ξ, E1 H)EH 1 = 0, we obtain (1.22) N ξ H = 1 2 N ξ N E H 1 E H 1 = 0. From (1.21) and (1.22), we see that H is parallel if and only if either γ is a geodesic in M n (c); or γ is a circle in M n (c) with the curvature κ 1 = 2 H = constant > 0. Obviously, in the first case, Σ 2 is a minimal surface. In the second case, the (2, 0)- part of Q vanishes if and only if that is equivalent to 16 H 4 + c H 2 + 3c ϕe H 1, H 2 = 0, 4κ 2 1 + c(1 + 3τ 2 12) = 0. Proposition 1.22 ([77]). A vertical cylinder Σ 2 = π 1 (γ) in M n (c) R has non-zero parallel mean curvature vector and the (2, 0)-part of the quadratic form Q vanishes on Σ 2 if and only if c < 0 and the curve γ is a circle in M n (c) with curvature κ = (1/2) c(1 + 3τ 2 ), where τ is the complex torsion of γ.

Reduction of Codimension and Holomorphic Differentials 15 Remark 1.23. S. Maeda and T. Adachi proved in [102] that for any positive number κ and for any number τ, such that τ < 1, there exists a circle with curvature κ and complex torsion τ in any complex space form. Therefore, for any c < 0, we know that circles γ, as in the previous proposition, do exist. Since 0 τ 2 1, we get that c/2 κ c, which means that the mean curvature of a non-minimal pmc cylinder Σ 2 = π 1 (γ), with vanishing (2, 0)-part of Q, satisfies c/4 H c/2. 3.4. Reduction of codimension. Let Σ 2 be a non-minimal pmc surface in a non-flat cosymplectic space form N 2n+1 (c), n 2. From Ricci equation (1.5) and the expression (1.20) of the curvature tensor R N, we get the following lemma. Lemma 1.24. For any vector field V normal to Σ 2, which is also orthogonal to ϕ(t Σ 2 ) and to ϕh, we have [A H, A V ] = 0, i.e., A H commutes with A V. Corollary 1.25. At each point p Σ 2, either H is an umbilical direction or there exists a basis that diagonalizes simultaneously A H and A V, for all normal vectors V satisfying V ϕ(t Σ 2 ) and V ϕh. In the following, we will study the case when our surface Σ 2 is pseudo-umbilical. Since H is also parallel, we have R N (X, Y )H = N X N Y H N Y N XH N [X,Y ] H = H 2 ( N XY N Y X [X, Y ]) = 0, for any tangent vector fields X and Y. We shall prove that, in this case, ξ T Σ 2 and ϕ(t Σ 2 ) NΣ 2, where NΣ 2 is the normal bundle of the surface. First, we have the following lemma. Lemma 1.26 ([77]). If Σ 2 is a pseudo-umbilical pmc surface, then the following four relations are equivalent: (i) ξ T Σ 2 ; (ii) H ξ; (iii) ϕ(t Σ 2 ) NΣ 2 ; (iv) ϕh T Σ 2. Proof. Let us again consider isothermal coordinates (u, v) on the surface and complex vector fields Z and Z as in the proof of Theorem 1.4. Since H is umbilical, we have σ(z, Z), H = 0 and then Q(Z, Z) = c H 2 (η(z)) 2 + 3c ϕz, H 2. Since Q(Z, Z) is holomorphic and H is umbilical and parallel, it follows that Z, Z η(z)η(h) + 3 ϕz, H ϕz, Z = 0. Now, it is easy to see that η(z)η(h) = 0 is equivalent to ϕz, H ϕz, Z = 0, and then we only have to prove the equivalence between (i) and (ii) and between (iii) and (iv), respectively. First, we prove that (i) is equivalent to (ii). If η(z) = 0 then 0 = η( N Z Z) = Z, Z η(h), since N 2n+1 (c) is a cosymplectic space form and N Z Z = Z, Z H. Conversely, if η(h) = 0, we have η( N Z H) = η(a H Z) = H 2 η(z) = 0.