SADAHIRO MAEDA Communicated by Takumi Yamada (Received: March 7, 2019) A BSTRACT . In a complex n(≥ 2)-dimensional complex projective space CP n (c) of constant holomorphic sectional curvature c(> 0), a type (A) hypersurface is one of fundamental examples in the theory of real hypersurfaces isometrically immersed into this ambient space. The purpose of this paper is to make a survey of fundamental properties of type (A) hypersurfaces in CPn(c).
1. I NTRODUCTION
We first recall the classification theorem of homogeneous real hypersurfaces M 2n-1 in CPn (c),n ≥ 2, that is they are orbits of some subgroups of the full isometry group I(CPn(c))(= SU(n + 1)). By virtue of the results in [32, 25, 12] we obtain the following. In CPn(c) ( n ≥ 2), a homogeneous real hypersurface is locally congruent to one of the following Hopf hypersurfaces all of whose principal curvatures are constant: (A 1 ) A geodesic sphere of radius r , where 0 < r < π/ c ; (A 2 ) A tube of radius r around a totally geodesic CP t (c) (1 ≤ l ≤ n − 2), where 0 < r < π/√c ; (B) A tube of radius r around a complex hyperquadric CQn-1, where 0 < r < π/(2√c ); (C) A tube of radius r around the Segre embedding of CP 1 (c) × CP(n-1)/2 (c), where 0 < r < π/(2√c ) and n (≥ 5) is odd; (D) A tube of radius r around the Plüker embedding of a complex Grassmannian CG 2.5, where 0 < r < π/(2√c ) and n = 9; (E)  A tube of radius r around a Hermitian symmetric space SO(10)/U(5), where 0 < r < π/(2√c ) and n = 15. These real hypersurfaces are said to be of types (A 1 ), (A 2 ), (B), (C), (D) and (E). Unifying real hypersurfaces of types (A 1 ) and (A 2 ), we call them type (A) hyper- surfaces. The numbers of distinct principal curvatures of these real hypersurfaces are 2, 3, 3, 5, 5, 5, respectively (for details, see [33]). We next recall the following fact (see [18]): Let M 2n −1 be a real hypersurface in CPn(c),n ≥ 2.  Then the length of the derivative of the shape operator A of M satisfies ||∇ A ||² ≥ (c²/4)(n – 1) at its each point. In particular, ||∇ A ||² = (c²/4)(n − 1) holds on M if and only if M is locally congruent to a type (A) hypersurface.  So it is natural to pay attention to type (A) hypersurfaces in the class of all real hypersurfaces in CPn (c). On the other hand, CPn(c) does not admit totally umbilic real hypersurfaces M 2n-1 . Hence, there exist no real hypersurfaces all of whose geodesics are mapped to circles in this space. So, in some sense the geometry of real hypersurfaces in CPn(c) is a bit complicated. The purpose of this paper is to survey geometric properties of type (A) hyper- surfaces M by observing the extrinsic shape of geodesics on M from the ambient space CP n ( c ). In particular, we investigate hypersurfaces of type (A 1 ) in detail. Needless to say, there do exist similarly real hypersurfaces isometrically immersed into a complex n(≥ 2)-dimensional complex hyperbolic space CH n (c) of constant holomorphic sectional curvature c(< 0), which are called type (A) hypersurfaces (for details, see [25]).  There are some analogous results to our Theorem 1, Theorem 2, Theorem 3, Theorem 5, Theorem 6, Theorem 7 and Theorem 10. However we emphasize that analogous results to our Theorem 4, Theorem 8 and Theorem 9 do not hold.
2. T ERMINOLOGIES AND FUNDAMENTAL RESULTS ON REAL HYPERSURFACES
Let M²n−1 be a real hypersurface with unit normal vector field N of an n (≥ 2)- dimensional complex projective space CPn (c) of constant holomorphic sectional curvature c(> 0).  The Riemannian connections V of CPn(c) and V of M are related by the following: (2.1) ∇ x Y = ∇ x Y + g(AX,Y)N  and   ∇ x N = −AX for all vector fields X and Y on M , where g denotes the metric induced from the standard Riemannian metric of CP n (c) and A is the shape operator of M in CP n (c) associated with N . On M an almost contact metric structure (φ,ξ,η,g) associated with N is canonically induced from the Kähler structure ( g, J ) of the ambient space CP n (c). They are defined by g(φX,Y) = g(JX,Y), ξ = −JN   and   η(X) = g(ξ,X) = g(JX,N). Note that by changing N for − N we have two almost contact metric structures (φ, ξ, η, g) and (φ, −ξ, −η, g) on M . It follows from (2.1) and the property ∇ J = 0 that (2.2) ∇ xξ = φAX   and   (∇ xφ)Y = η(Y)AX − g(AX,Y)ξ   for each X,Y ∈ TM. The above equations do not depend on the choice of the unit normal vector N . We denote by R the curvature tensor of M . Then R is given by (2.3) g((R(X,Y)Z,W) = (c/4){g(Y, Z)g(X, W) − g(X, Z)g(Y, W) + g(φY, Z)g(φX, W) − g(φX, Z)g(φY, W) − 2g(φX, Y)g(φZ, W)} + g(AY, Z)g(AX, W) – g(AX, Z)g(AY, W). The following is called the equation of Codazzi. (∇ x A)Y − (∇ y A)X = (c/4)(η(X)φY − η(Y)φX − 2g(φX,Y)ξ). Let K be the sectional curvature of M . That is, K is defined by K(X,Y) = g(R(X,Y)Y,X), where X and Y are orthonormal vectors on M . Then it follows from (2.3) that K(X,Y) = (c/4)(1 + 3g($X,Y)²) + g(AX,X)g(AY,Y) − g(AX,Y)². We call eigenvalues and eigenvectors of the shape operator A principal curva- tures and principal curvature vectors of M in CPn (c), respectively.  Here and in the following, we set V λ := {X ∈ TM | AX = λ X}. We usually call M a Hopf hypersurface if the characteristic vector ξ of M is a principal curvature vector at each point of M .
3. CIRCLES IN RIEMANNIAN GEOMETRY
The notion of circles is a key in this paper.  A smooth real curve γ = γ(s) parametrized by its arclength s on a Riemannian manifold M with Riemannian connection ∇ is called a circle of curvature k if it satisfies ordinary equations
along the curve γ, where k is a positive constant and Y, is the principal normal unit vector perpendicular to γ(s). A geodesic is regarded as a circle of null curvature. By virtue of the unique existence theorem on ordinary equations for given each positive k and each pair of orthonormal vectors X and Y at an arbitrary point p of M  there exists locally the unique circle γ = γ(s) of curvature k satisfying the initial Riemannian manifold every circle can be defined for −∞ < s < ∞ (see [27]). In this paper, we consider circles in CPn(c),n ≥ 2.  Given a circle γ = γ(s) satisfying (3.1) we call τ(s) := g( i (s), JY s ) the holomorphic torsion of γ , where J is the standard complex structure of CP n (c). The function −1 ≤ τ ≤ 1 is constant along every circle γ in CPn(c). In fact,
It is well-known that all geodesics on CPn (c) are congruent to each other by some ϕ ∈ I(CPn(c)). The congruence theorem on circles of positive curvature in CPn(c) is expressed as: Lemma 1 ([6]). Let γ i = γ i (s) (i = 1,2) be circles of positive curvature k i and holomorphic torsion τ i in CPn(c),n ≥ 2.  Then these two circles γ and γ2 are congruent to each other if and only if k 1 = k 2 and [ 7 ] = [ 7 ].  Precisely, when T1 = T2 (resp. T1 = −72), they are congruent to each other by some holomorphic isometry (resp. anti-holomorphic isometry) in this space. A circle γ of the holomorphic torsion τ with τ = 1 or τ = −1 (resp. τ = 0) is called a Kähler circle (resp. totally real circle). Note in CPn(c) that a circle γ is totally real if and only if γ lies locally on a totally real totally geodesic surface RP²(c/4) of constant sectional curvature c/4 and a circle γ is Kähler if and only if γ lies locally on a totally geodesic holomorphic line CP 1 (c). They are closed curves. But, in general a circle with holomorphic torsion τ̸= 0, ±1 is not necessarily closed (see [6, 3]).
4. FRENET CURVES
In general, M n denotes a real n-dimensional Riemannian manifold.  In this paper, M n denotes a complex n -dimensional Kähler manifold. A smooth curve γ = γ(s) parametrized by its arclength s on a complex n- dimensional Kähler manifold Mn furnished with Riemannian metric g and Rie- mannian connection ∇ is s said to be a Frenet curve of proper order d (2 ≤ d ≤ 2n) if there exist an orthonormal system {V 1 = ÿ, V 2 , V 3 , . . , V a } of vector fields along γ and positive smooth functions κ 1 (s), . . , κ d − 1 (s) satisfying the following system of ordinary differential equations: ∇ γ V j (s) = − κ j − 1 (s)V j − 1 (s) + κ j (s)V j +1 (s), 1 ≤ j ≤ d. Here, κ 0 V 0 and κ d V d+1 are null vector fields along γ .  The functions κ 1 , . . , κ d −1 and the orthonormal frames {V 1 ,V 2 ,...,V a} are called the curvatures and the Frenet frame of the curve γ, respectively.  Roughly speaking, a Frenet curve is a smooth curve having no inflection points of higher order. For the Frenet frame {V 1 , V 2 , . . , V a } of γ , we set τ ij (s) := g(V i (s), JV j (s)) with 1 ≤ i < j ≤ d and call them the holomorphic torsions along γ. A Frenet curve is s aid to be a helix when all of its curvatures k 1 , . . , k d − 1 are constant functions. A helix of proper order 2 is a circle of curvature k ( = k 1 > 0). A real curve γ = γ(s) in M n is said to be homogeneous if it is an orbit of one-parameter subgroup of I( M n ). In the following, we adopt CPn(c), n ≥ 2 as a Kähler manifold Mn. In the study of Frenet curves in CPn (c), the notion of holomorphic torsions plays an important role.  We can give the necessary and sufficient condition for a Frenet curve to be homogeneous in CPn (c) by using the notion of curvatures and holomorphic torsions. In fact, the following is known. Proposition 1 ([21]). A real curve γ = γ(s) is a homogeneous curve in CPn(c), n ≥ 2 if and only if it is a helix and al of its holomorphic torsions are constant func- tions. For a circle γ of positive curvature k in CP n (c), we have just one holomorphic torsion τ(s) := τ12(s) = g(V1(s), JV2(s)).  By easy computation in the previous section we find that the holomorphic torsion τ of a circle γ of positive curvature is automatically a constant function.  Hence, by Proposition 1 we see that every circle of positive curvature is homogeneous in CPn (c). This, together with a fact that all geodesics are homogeneous curves in CPn(c), implies that all circles are homogeneous in this space.  Hence every circle in CP n ( c ) is an integral curve of some (Killing) vector field, which implies that it is a simple curve in this space.
5. C ONGRUENCE THEOREM ON GEODESICS OF TYPE (A) HYPERSURFACES
We first review the following: Lemma 2 ([29, 18]). Let M be a real hypersurface isometrically immersed into CPn(c),n ≥ 2.  Then the following four conditions are mutually equivalent: (1) M is locally congruent to a type (A) hypersurface ; (2) The shape operator A of M satisfies (∇ x A)Y = −(c/4)(g(φX,Y)ξ + η(Y)φX)   for X,Y ∈ TM; (3) The shape operator A' of a hypersurface π− 1 (M) in a Euclidean sphere S 2n+1 (c/4) of constant sectional curvature c/4 is parallel, where π : S²n+¹(c/4) → CP n (c) is the Hopf fibration; (4) The structure tensor φ and the shape operator A of M satisfy φA = Aφ on M. Remark 1. We explain principal curvatures of type (A) hypersurfaces M in CP n (c) with n ≥ 2. It is well-known that if M is of type (A 1 ), M has two distinct constant principal curvatures δ = √c cot(√c r) with multiplicity 1 and λ = (√c/2). cot( √ c r/2) with multiplicity 2n – 2, and that if M is of type (A 2 ), M has three distinct constant principal curvatures δ = √ c cot(√ c r ) with multiplicity 1, λ 1 = (√c/2) cot(√c r/2) with multiplicity 2n − 2l − 2 and λ 2 = −(√c/2) tan(√c r/2) with multiplicity 2 ℓ , where Aξ = √c cot(√c r)ξ and δ can be expressed as: δ = λ 1 + λ 2 (cf. [25]). Remark 2. We review another expression of type (A) hypersurfaces M in CP n (c) with n ≥ 2. We set M ′ = π− 1 M . By Lemma 2(3) we know that M ′ is a Clif- ford hypersurface M 2p+1,2t+1(c 1 , c 2 ) := S 2p+1(c 1 ) × S 2t+1(c 2 ) in the ambient sphere S 2n+1 (c/4), where p, l are integers with p + l = n − 1 and p ≥ l ≥ 0 except p = ℓ = 0, and c 1 , c 2 are positive constants with 1/c 1 + 1/c 2 = 4/c. As a matter of course c 1 , c 2 and c/4 are sectional curvatures of these spheres. This hypersurface M 2p +1,2 ℓ+1( c 1 , c 2 ) has two distinct constant principal curvatures c 1 /√c 1 + c 2  with multiplicity 2p + 1 and − c 2 /√c 1 + c 2  with multiplicity 2 ℓ + 1.  Then the real hy- persurface M p ℓ := π( M 2p +1,2ℓ+1( c 1 , c 2 )) with p ℓ̸= 0 in CP n ( c ) has three constant principal curvatures (c 1 − c 2 )/√c 1 + c 2  with multiplicity 1 which is the principal curvature of the characteristic vector ξ on M P,ξ, c 1 /√c 1 + c 2  with multiplicity 2p and −c 2 / √c 1 + c 2  with multiplicity 2 l (for details, see [18]).  Note that the hy- persurface M p ℓ = π( M 2 p +1,2 ℓ+1( c 1 , c 2 )) is either a hypersurface of type (A 1 ) or a hypersurface of type (A 2 ) in the introduction when ℓ = 0 or ℓ > 0, respectively. Hence, the radius r of a type (A) hypersurface must satisfy cot(√c r/2) = √c 1 /c 2 . Next, let γ = γ(s) be a geodesic parametrized by its arclength s on a type (A) hypersurface M ² n − 1 in CP n (c), n ≥ 2. We consider two functions ρ γ = ρ γ (s) and κ γ = κ γ (s) along the curve γ defined by ρ γ (s) := g( γ (s),ξ γ (s)) and κ γ (s) := g(Aj(s), j(s)). Then by the first equality in (2.2) and the skew-symmetry of φ : g(φX,Y) = −g(X,φY) and Lemma 2 we see that these functions ρ ~ and κ ~ are constant along each geodesic γ on every type (A) hypersurface. We call ρ λ and κ λ of a geodesic γ on a type (A) hypersurface M the strcture torsion and the normal curvature of the curve γ , respectively. Using these two invariants ρ γ and κ γ of a geodesic γ , we can describe the following congruence theorem on geodesics: Lemma 3 ([4]). Let γ i (i = 1,2) be geodesics on a type (A) hypersurface M in CPn(c), n ≥ 2.  Then the following hold: (A 1 ) When M is of type (A 1 ), the structure torsions of these curves satisfy | ργ | = |ρ ~ ] if and only if they are congruent to each other, i.e., there exists an isometry on M with γ2(s + so) = (ϕ σγ1)(s) for each s and some so. (A 2 ) When M is of type (A 2 ), the structure torsions and normal curvatures of these curves satisfy | ργι | = | ργι | and κγι = κγι if and only if they are congruent to each other. Given a submanifold M n isometrically immersed through f into a Riemannian manifold M n+ P we call a smooth curve γ = γ(s) an extrinsic geodesic on M n if the curve f o γ is a geodesic in M n+ρ . Needless to say, every extrinsic geodesic is also a geodesic on the submanifold. Here, using Lemma 3, we count the number of congruent classes of extrinsic geodesics on every type (A) hypersurface M with respect to the full isometry group I(M) of M. Proposition 2. Let M be a type (A) hypersurface of radius r (0 < r < π/√c ) of CPn(c), n ≥ 2.  Then the following hold: (1) When M is of type (A 1 ) of radius r (0 < r < π/(2 √ c )), M has no extrinsic geodesics. (2) When M is of type (A1) of radius r (π/(2√c) ≤ r < π/√c), M has just one congruent class of extrinsic geodesics. (3) When M is of type (A 2 ) of radius r (0 < r < π/√√), M has uncountably infinite congruent classes of extrinsic geodesics. Proof. The proofs of Statements (1) and (2) are given in the proof of Theorem 1 in [30]. It remains to show Statement (3). For an extrinsic geodesic γ = γ(s) on M of type (A 2 ) we set γ(0) = ργξγ(0) + αX + bY , where ργ, α, b are nonnegative with ρ 2 , + α 2 + δ 2 = 1 and X,Y are unit vectors with X ∈ V(√ ε/2)cot(√ ε r/2) and Y ∈ V − (√ ε/2) tan(√ r/2) .  By the constancy of the normal curvature κ γ of the geodesic γ we see that γ is an extrinsic geodesic if and only if g(A γ (0), γ (0)) = 0. Then by simple computation we have ρ 2 = tan 2 (√ c r/2) − a 2 sec 2 (√ c r/2). This, together with ρ 2 , ≤ 1, yields inequalities sin 2 (√ c r/2) − cos 2 (√ c r/2) ≤ α 2 ≤ sin 2 (√ c r/2). So we get a one-parameter family γα = γα(s) of extrinsic geodesics on our type (A) hypersurface. Hence we obtain the desirable conclusion (see Lemma 3((A2))).   □ As an immediate consequence of the proof in Proposition 2(3) we have Remark 3 ([30]). When M is of type (A 2 ) of radius π/(2 √ c ), M has a one- parameter family of extrinsic geodesics γα = γα ( s ) with 0 ≤ α ≤ 1 /√ 2 . The initial vector γ(0) is written as: γ(0) = √ 1 − 2α² ξγ(0) + αX + αY , where X and Y are unit vectors with X ∈ V f ∈ /2 and Y ∈ V − /ε /2 . Note that two curves γα and γ b are congruent to each other with respect to I( M ) if and only if a = b ∈ [0,1/√2]. On the contrary these curves are congruent to each other with respect to the isometry group SU( n + 1) of the ambient space CP n ( c ) for any a, b ∈ [0, 1/√2]. 6. E XTRINSIC SHAPE OF GEODESICS ON TYPE A HYPERSURFACES It is known that every geodesic γ = γ(s) on a type (A) hypersurface M is a homogeneous curve (see [24]). This, together with a fact that our real hypersurface M is homogeneous in CPn(c), the curve γ is also mapped to a homogeneous curve in this ambient space.  Hence the curve γ is a helix in CPn(c), so that it has all constant curvatures k 1 , . . . , k d –1 . Motivated by this fact, we give the following characterization of type (A) hypersurfaces. Theorem 1 ([17]). Let M 2n-1 be a connected real hypersurface isometrically im- mersed into CPn(c), n ≥ 2. Then M is locally congruent to a type (A) hypersurface if and only if every geodesic γ of M, considered as a curve in the ambient space CPn(c), has constant first curvature k1(:= ||∇ 2 7 ||) along 7 , where ∇ is the Rie- mannian connection on CPn(c). The following is fundamental. Lemma 4. Let M n be a connected hypersurface isometrically immersed into a Riemannian manifold Mn+1. Then the following are equivalent: (1) The shape operator A of M n in M n+1 is expressed as A = λ n , where λ is a constant function on M n and I n is a unit matrix; (2) Every geodesic γ on M n , considered as a curve in the ambient space M n+1, is a circle; (3) Every geodesic γ on M n , considered as a curve in the ambient space M n+1, is a circle of the same curvature |λ|, where λ is given in Condition (1). By virtue of Lemma 4 and a fact that there exist no totally umbilic real hyper- surfaces in CP n (c) we find that CP n (c) admits no real hypersurfaces M 2n−1 all of whose geodesics are mapped to circles in this ambient space. We now pose the following problem: Problem 1. Does there exist real hypersurfaces M 2 n − 1 some of whose geodesics are mapped to circles of the same curvature in CPn(c), n ≥ 2 through an isometric immersion? The following is a partial answer to Problem 1. Fact 1 ([20]). (1) Every geodesic γ = γ(s) on M²n−1 of type (A 1 ) of radius r (0 < r < π/√c ) with initial vector γ(0) perpendicular to the characteristic vector ξγ(0) is mapped to a circle of the same positive curvature (√c /2) cot(√c r/2) in CPn(c). (2) For a geodesic γ = γ(s) on M 2n−1 of type (A 2 ) the following hold: (2i) When the initial vector γ(0) is a principal curvature vector of M²n−1 in CPn(c) with principal curvature (√c/2) cot(√c r/2), the curve γ is mapped to a circle of the same positive curvature (√c/2) cot(√c r/2) in the ambient space; (2ii) When the initial vector γ(0) is a principal curvature vector of M²n−1 in CPn(c) with principal curvature −(√c/2) tan(√c r/2), the curve γ is mapped to a circle of the same positive curvature (√c /2) tan(√c r/2) in the ambient space. Considering the converse of Fact 1, we obtain the following (see Proposition 2): Theorem 2 ([20, 30]). Let M 2n-1 be a connected real hypersurface isometrically im- mersed into CPn(c), n ≥ 2. Suppose that there exist orthonormal vectors v1, v2, . . . , v2n−2 orthogonal to the characteristic vector ξρ at each fixed point p of M such that all geodesics γ i = γ i (s) on M  with γ i (0) = p and γ i (0) = ν i (1 ≤ i ≤ 2n − 2) are mapped to circles of the same positive curvature k(p) in CPn(c).  Then the function k = k (p) is locally constant on M and M is locally congruent to either a geodesic sphere G(r) of radius r (0 < r < π/√c ) or a hypersurface of type (A2) of radius π/(2√c). Moreover, the following hold: (1) When M has no extrinsic geodesics, M is locally congruent to a geodesic sphere G(r) of radius r = (2/√c)cot-1(2k/√c) (0 < r < π/(2√c)) in CPn(c) with k > <c/2; (2) When M has just one congruent class of extrinsic geodesics with respect to isometry group I(M) of M , M is locally congruent to a geodesic sphere G(r) of radius r = (2/√c)cot-1(2k/√c) (π/(2√c) ≤ r < π/√c) in CPn(c) with k ≤ √c/2; (3) When M has at least two congruent classes of extrinsic geodesics with re- spect to I(M) of M, M is locally congruent to a hypersurface of type (A2) of radius π/(2√c) in CPn(c) with k = √c/2. In the following, we shall characterize all homogeneous real hypersurfaces M 2n–1 in CPn(c),n ≥ 2 by observing the extrinsic shape of geodesics from this ambient space. For this purpose we prepare the following two lemmas: Lemma 5. Let Mn be a connected hypersurface isometrically immersed into a Riemannian manifold Mn+1.  If a geodesic γ = γ(s) (s ∈ I ⊂ R) on M is mapped to a circle of positive curvature k , then the shape operator A of M n in M n+1 satisfies either Ay(s) = ky(s) for all s ∈ I or Ay(s) = −ky(s) for all s ∈ I. Lemma 6 ([2]). Let γ = γ(s) be a geodesic on each homogeneous real hypersurface M²n−1  in CPn(c),n ≥ 2.  If the initial vector γ(0) is a principal curvature unit vector with principal curvature (, say) λ, then the curve γ is mapped to a circle of curvature |λ| in CPn(c). In particular, when the initial vector i(0) is perpendicular to ξγ(0), the curvature |λ| is positive. In view of Lemmas 5, 6 and the classification theorem of all homogeneous real hypersurfaces in CPn (c) we obtain the following: Theorem 3 ([2]). Let M 2n-1 be a connected real hypersurface isometrically im- mersed into CPn(c),n ≥ 2.  Then M  is locally congruent to a homogeneous real hypersurface if and only if there exist orthonormal vectors v 1 , . . . , v 2n –2  orthogo- nal to the characteristic vector ξ p at each point p of M such that all geodesics γi = γi(s) (1 ≤ i ≤ 2n − 2) on M  with γi(0) = p  and γi(0) = vi  are mapped to circles of positive curvature in CPn(c). Note that all circles in Theorem 3 are totally real circles in the ambient space (CPn(c). 7. R EAL HYPERSURFACES HAVING φ -INVARIANT SHAPE OPERATOR We first recall the following: Proposition 3 ([11]). Let ( Mn, g, J ) be a complex n-dimensional Kähler manifold immersed into a (2n + p)-dimensional sphere S 2n+p(c) of constant sectional curva- ture c through an isometric immersion f . Then f has parallel second fundamental form σ if and only if σ is J-invariant, namely σ(JX, JY) = σ(X,Y) holds for all vectors X,Y  on Mn. Remark 4. The "if" part in Proposition 3 is obtained by tensor calculus.  The "only if" part in this proposition is based on a fact that under the only if part our Kähler manifold M n is locally congruent to a compact Hermitian symmetric space and moreover this isometric parallel immersion f of the compact Hermitian symmetric space into the ambient sphere S 2 n+p (c) is locally realized as a part of the embedding as the symmetric R-space (cf. [11, 15, 26]). Inspired by Proposition 3, for each real hypersurface M ² n − 1 furnished with al- most contact metric structure (φ,ξ,η, g) isometrically immersed into CPn(c), n ≥ 2 we introduce the following conditions concerning φ -invariances of the shape oper- ator A of M in the ambient space CPn(c). The shape operator A of M is called strongly φ-invariant if A satisfies (7.1) g(AφX,φY) = g(AX,Y), i.e., σ(φX,φY) = σ(X,Y) for all vectors X and Y on M . Also, it is called weakly φ -invariant if A satisfies (7.2) g(AφX,φY) = g(AX,Y), i.e., σ(φX,φY) = σ(X,Y) for all vectors X and Y orthogonal to the characteristic vector ξ on M . The following is a classification theorem of real hypersurfaces in CPn(c) having strongly φ -invariant shape operator. Theorem 4 ([19]). Let M 2 n-1 be a connected real hypersurface isometrically im- mersed into CPn(c),n ≥ 2.  Then the following conditions (1),(2) and (3) are mutually equivalent. (1) M is locally congruent to a type (A) hypersurface with radius π/(2√c). (2) The shape operator A of M in CPn(c) is strongly φ-invariant. (3) M satisfies the following two conditions: (3i) At each fixed point p ∈ M , there exist orthonormal vectors v 1 , v 2 , . . . , v 2n − 2 orthogonal to the characteristic vector ξρ of M such that all geodesics γ i ＝ γ i (s) on M with γ i (0) = p and γ i (0) = ν i (1 ≤ i ≤ 2n − 2) are mapped to circles of the same positive curvature in CPn(c); (3ii) There exists at least one integral curve of the characteristic vector field ξ of M which is an extrinsic geodesic on M . Remark 5. Theorem 4(3) gives a geometric meaning of a condition that M has strongly φ -invariant shape operator in the ambient space CPn(c). Next, we classify Hopf hypersurfaces having weakly φ -invariant shape operator in CPn(c). Proposition 4 ([19]). For a real hypersurface M²n-1 isometrically immersed into CPn(c), n ≥ 2 the following two conditions are mutually equivalent. (1) M is a Hopf hypersurface having weakly φ-invariant shape operator. (2) M is locally congruent to a type (A) hypersurface. Remark 6. There exist many non-Hopf hypersurfaces M 2 n −1 having weakly φ - invariant shape operator in CPn(c) (for details, see [19]). 8. LENGTH SPECTRUM OF HYPERSURFACES OF TYPE (A 1 ) IN CP n (c) Having respect for [8], we call a compact simply connected Riemannian manifold a Berger sphere if its sectional curvatures lie in the interval [ δ K, K ] with some positive constants K and δ ∈ (0,1/9) and has a closed geodesic of length shorter than 2π/√K. When M is even dimensional, Klingenberg ([13]) showed that on a compact simply connected Riemannian manifold whose sectional curvatures lie in (0, K ] with some constant K every geodesic on M has length not shorter than 2π/√K. Thus, we have only odd dimensional Berger spheres. Weinstein ([36]) gave examples of Berger spheres by observing geodesic spheres G(r) of the radius r (0 < r < π/√c ) with tan²(√c r/2) > 2 in CPn(c). We find easily the following: Lemma 7. Let G(r) be a geodesic sphere of radius r (0 < r < π/√c ) in CPn(c), n ≥ 2. Then the following three conditions are mutually equivalent: (1) The radius r satisfies an inequality tan²(√c r/2) > 2. (2) The sectional curvature K of G(r) satisfies sharp inequalities δ L ≤ K ≤ L for some δ ∈ (0,1/9) at its each point, where L = c + (c/4) cot 2 (√c r/2). (3) The length of every integral curve of the characteristic vector field ξ on G(r) is shorter than 2π/√L, where L is the maximal sectional curvature of G(r) which is given by Condition (2). Remark 7. For every geodesic sphere G( r ) (0 < r < π/√c ) in CPn ( c ) every integral curve of the characteristic vector field ξ on G ( r ) is a geodesic. In the following, we call a geodesic sphere G(r) (0 < r < π/√c) with tan²(√c r/2) > 2 in CPn (c) a Berger sphere. Motivated by their works, we shall investigate the distribution of lengths of closed geodesics on every hypersurface of type (A 1 ) in CP n (c), n ≥ 2. We first review fundamental notion on geodesics. 10 A smooth curve γ : R → M on a Riemannian manifold M parameterized by its arclength s is said to be closed if there is a positive sc satisfying γ(s + sc) = γ(s) for all s. When γ is closed, the minimum positive such s c is called its length and is denoted by length( γ ). When γ is open, that is, it is not closed, we set length( γ ) = 00. We here review the congruency for smooth real curves. We say two smooth curvesﾌﾘ,ﾘ2 on M parameterized by its arclength s to be congruent to each other if there exist an isometry ϕ of M and a constant s 0 satisfying γ 2 (s) = φ o γ 1 (s + s 0 ) for all s. When we can take s 0 = 0, we say that they are congruent to each other in strong sense. We denote by G ( M ) the moduli space of geodesics of unit speed on M , which is the set of all congruent classes of geodesics. We can then define a map L : G(M) → (0,00] by L([γ]) = length(γ), where [γ] denotes the congruent class containing a geodesic γ. This map or sometimes its image L(G) ∩ (0, ∞) on the real line is called the length spectrum of M. The aim of this section is to study how lengths of closed geodesics on geodesic spheres G( r ) (0 < r < π/√c ) of CPn(c ) are distributed on the real line.  To do this we define functions m G(r), n G(r) : (0, ∞) → N U {∞}, where N is the set of positive integers, as follows: For a positive number λ, we denote by mG(r)(λ) the number of congruent classes of geodesics of length λ, and denote by n G(r)(λ) the number of congruent classes of geodesics of length not longer than λ . We call mG(r)(λ) the multiplicity of lengths at λ. We first have Theorem 5 ([7]). Given a geodesic γ on a geodesic sphere G(r) of radius r (0 < r < π/√c) in CPn(c),n ≥ 2 through a natural isometric embedding ι we find the following. (1) The curve ι σ γ is either a geodesic, a Kähler circle, a totally real circle or a homogeneous curve of proper order 4 in the ambient space CPn(c); (2) If the structure torsion of γ is ±1, then γ is closed and its length is (2π/√c)sin(√cr); (3) If γ has null structure torsion, then γ is also closed and its length is (4π/√c)sin(√c r/2); (4) When the structure torsion of γ is of the form sin θ (0 < |θ| < π/2), it is closed if and only if
with some relatively prime positive integers p and q with q < p tan²(√c r/2). In this case, its length is
As a direct consequence of Theorem 5, for a geodesic sphere G( r ) of radius r in CPn(4), we can see that Lspec(G(r)) = {π sin 2r} ∪ {2π sin r}
Therefore we obtain the following Theorem 6 ([7]). On a geodesic sphere G(r) (0 < r < π/√c) in CPn(c), there exist countably infinite congruent classes of closed geodesics. Moreover the length spectrum Lspec(G(r)) of G(r) is a discrete unbounded subset in the real line R. Moreover, we establish the following Theorem 7 ([7]). Geodesics on a geodesic sphere G(r) of radius r (0 < r < π/√c ) in CPn(c) with n ≥ 2 satisfy the following properties. (1) When tan²(√c r/2) is irrational, two closed geodesics on G(r) are congruent to each other in strong sense if and only if they have a common length. (2) When tan²(√c r/2) is rational, the multiplicity mc(r)(λ) at λ is finite at each positive λ, but is not uniformly bounded. It satisfies lim sup →∞ m(λ) = ∞ and lim λ→∞ λ−δ m(λ) = 0 for each δ > 0. (3) We have
In the rest of this section we have the following characterizations of Berger spheres from the viewpoint of submanifold theory. Theorem 8 ([14]). Let M 2n-1 be a connected real hypersurface of CPn(c), n ≥ 2 through an isometric immersion. Then M is locally congruent to a Berger sphere, namely a geodesic sphere G(r) of radius r with tan²(√cr/2) > 2, with respect to the full isometry group SU(n + 1) of the ambient space CPn(c) if and only if at each point p of M  there exists an orthonormal basis v 1 , . . . , v 2n – 2 , ξ p of T p M such that all geodesics γ i = γ i (s) (1 ≤ i ≤ 2n − 2) with initial condition that γί(0) = p and γί(0) = vi are mapped to circles of the same positive curvature k(p) with k(p) < √ C/(2√ 2 ) in the ambient space CP n (c), where ξ p is the characteristic vector of M at p ∈ M . In this case, the function k = k ( p ) on M is automatically constant with k = (√c/2) cot(√c r/2). Theorem 9 ([14]). Let M 2n-1 be a connected real hypersurface of CP n (c), n ≥ 2 through an isometric immersion. Then M is locally congruent to a Berger sphere if and only if M  satisfies the following two conditions. (1) There exists a positive constant k with k < √ C/(2√ 2 ) such that the exterior derivative dη of the contact form η on M satisfies either dη(X,Y) = kg(φX,Y) for all X,Y ∈ TM  or dη(X,Y) = −kg(φX,Y) for all X,Y ∈ TM, where g and φ are the Riemannian metric and the structure tensor on M , respectively. (2) There exists a point x of M satisfying that every sectional curvature of M at x is positive. The definition of dη on a real hypersurface M is given by dη( X, Y ) = (1/2). {X(η(Υ)) − Υ(η(Х)) − η([Х,Υ])} for all Х,Υ ∈ ТМ. We note for comparison the recent paper of Li, Vrancken and Wang ([16]), which gives a characterization of 3-dimensional Bereger spheres as Lagrangian submani- folds of CP3. They showed the following (for details, see Theorem 1.2 in [16]): Let φ be a Lagrangian isometric immersion of a homogeneous 3-manifold M 3 into a complex space form M 3 (c) (= CP 3 (c), CH 3 (c) or C 3 ). Then c > 0 and φ is minimal and M 3 is locally congruent to the Berger sphere. At the end of this section we give some comments: Remark 8. (1) Except geodesics with structure torsion ±1, every geodesic γ on G(r) (0 < r < π/√c ) in CPn(c) satisfies length(γ) > 4π/√c(4 + cot²(√c r/2)) , i.e., it satisfies an inequality of Klingenberg's type. (2) In the class of all homogeneous real hypersurfaces M 2n−1 in CPn(c), M is of type (A 1 ) if and only if every sectional curvature of M is positive, and M is of type (A) if and only if every sectional curvature of M is nonnegative (see [22]). (3) In the statement of Theorem 9, if we remove Condition (2), this theorem does not hold. The Berger sphere and a certain homogeneous real hypersurface of type (B) satisfy Theorem 9(1).  We here review a fact that CPn(c) admits no real hypersurfaces with dη = 0 (see [25]). On the other hand, a complex Euclidean space C n has real hypersurfaces M 2n−1 with dη = 0 (for example, the totally geodesic real hypersurface R 2n-1 satisfies this condition). So, in some sense the geometry of real hypersurfaces of CPn(c) is more complicated than that of Cn. Motivated by them, we establish Theorem 9.
9. ALMOST CONTACT STRUCTURES ON REAL HYPERSURFACES IN CPn(c)
We first clarify the meaning of the condition that a real hypersurface M in CPn(c),n ≥ 2 is a Sasakian manifold with respect to almost contact metric struc- tures (φ,ξ,η, g) and (φ, −ξ, −η, g). We call a real hypersurface M Sasakian if M satisfies either (∇ x φ )Y = g(X,Y)ξ − η(Y)X for all tangent vectors X,Y ∈ T p M at an arbitrary point p ∈ M or (∇ x φ )Y = −g(X,Y)ξ + η(Y)X for all vectors X,Y ∈ T p M at an arbitrary point p ∈ M , where ∇ denotes the Riemannian connection of M (see Theorem 6.3 in [9]). For a tangent vector X of a Sasakian manifold M which is orthogonal to ξ , the sectional curvature of the plane spanned by X and φ X is called φ -sectional curvature of the φ -section determined by X. We say a Sasakian manifold to be a Sasakian space form of constant φ -sectional curvature k if φ -sectional curvatures of all tangent vectors orthogonal to ξ are k. By easy computation we find the following Lemma 8 ([1]). For a connected real hypersurface M 2n-1  isometrically immersed into CPn(c),n ≥ 2 the following four conditions are mutually equivalent. (1) M is locally congruent to a geodesic sphere G(r) of radius r (0 < r < π/√c) with (√c/2) cot(√c r/2) = 1. (2) The shape operator A of M in the ambient space CPn(c) is expressed as either AX = −X + (c/4)η(X)ξ for each X ∈ TM or AX = X − (c/4)η(X)ξ for each X ∈ TM . That is, M is a member of totally η-umbilic real hypersurfaces in CPn(c). (3) M is a Sasakian manifold with respect to the almost contact metric structure (φ,ξ,η, g) induced from the standard Kähler structure (g, J) on CPn(c). (4) M is a Sasakian space form with respect to the almost contact metric structure (φ,ξ,η,g) induced from the standard Kähler structure (g,J) on CPn(c). In this case, M has constant φ-sectional curvature c+1. Remark 9. (1) It follows from a fact that every geodesic sphere G(r) (0 < r < π/√c ) in CPn(c) is diffeomorphic to a Euclidean sphere and Lemma 8 that our geodesic sphere G(r) (0 < r < π/√c ) of radius r with (√c /2) cot(√c r/2) = 1 is a complete simply connected Sasakian space form N(k)(:= N 2n −1(k)) of constant φ - sectional curvature k = c+1(> 1). Hence by virtue of the unique existence theorem on complete simply connected Sasakian space forms in [35] our geodesic sphere G( r ) with (√c/2) cot(√c r/2) = 1 is congruent to a standard example of Sasakian space forms having constant φ -sectional curvature greater than 1 constructed in page 114 of [9]. (2) In view of (2.3) and Lemma 8(2) we can write the curvature tensor R of a Sasakian space form N ( k ) of constant φ -sectional curvature k ( > 1) (cf. [28]). Using the results in [7] and an equality (√c/2) cot(√c r/2) = 1, we give the following list of the length spectrum LSpec(N(k)) = L(G(N(k))) ∩ R of a complete simply connected Sasakian space form N ( k ) with k > 1
p and q are relatively prime positive integers which satisfy pq is even and 4p < (k − 1)q p and q are relatively prime positive integers which satisfy pq is odd and 4p < (k − 1)q Geodesics on a complete simply connected Sasakian space form N(k) with k > 1 have the following properties. Proposition 5. Every geodesic on N(k) (k > 1) is homogeneous, that is, it is an orbit of some one-parameter subgroup of the isometry group I(N(k)). Hence it is a simple curve, i.e., it does not have self-intersections. Proposition 6 ([5]).  We have countably infinite congruent classes of closed geodesics on N(k) (k > 1). Moreover, we have the following. (1) Every N(k) (k > 9) is a Berger sphere. 14 (2) When k is irrational, two closed geodesics on N ( k ) are congruent to each other if and only if they have a common length. (3) When k is rational, we have m N (k) (λ) is finite for each positive λ , but is not uni- formly bounded; lim sup λ →∞ m N ( k ) = ∞ . The growth order of the function m N ( k ) is less than polynomial order.  More precisely, we have limｽ→∞ λ−δ m N(k)(λ) = 0 for each positive δ. Though the feature of the function m N ( k ) of multiplicities depends whether k is rational or irrational, the functions n N ( k ) of numbers of congruent classes of closed geodesics have a common property. Theorem 10 ([5]). The function n N ( k ) with k > 1 satisfies
For the definitions of functions m N (k) and n N (k) see Section 8. 10. A CERTAIN HOMOGENEOUS SUBMANIFOLD IN A SPHERE In this section we show that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold M having the following prop- erties: (1) M is diffeomorphic but not isometric to a Euclidean sphere. (2) M is a homogeneous submanifold with nonzero parallel mean curvature vector in the ambient sphere. (3) M is a Berger sphere. (4) M is a Sasakian space form of constant φ -sectional curvature. For this purpose we establish the following theorem. Theorem 11 ([23]). (I) For each of c > 0,n ≥ 2,N > n(n + 2) − 1 and č ≤ (n + 1)c/(2n), there exists a (2n − 1)-dimensional submanifold M²n−1  isometrically immersed into an N-dimensional sphere SN ( č ) of constant sectional curvature č, which has the following properties: (1) M is diffeomorphic but not isometric to a Euclidean sphere. (2) M is a homogeneous submanifold which has nonzero parallel mean curvature vector with respect to the normal connection in S N ( č ). (3) M is a Berger sphere. (4) When c = 8n+4, M  is a Sasakian space form of constant φ -sectional curvature 8n + 5. (II) For each of c > 0 and n ≥ 2, when N = n(n + 2) − 1, there exists also a (2n − 1)-dimensional submanifold M 2n−1 in an N-dimensional sphere S N ( č ) of constant sectional curvature c = (n + 1)c/(2n), which has the above properties (1), (2), (3), (4). We explain an idea in the proof of Theorem 11. We denote by ( M, ι M ) a real hypersurface M 2 n −1 of CP n (c) through an isometric immersion ι M : M → CP n (c). In the following, we regard a real hypersurface M in CPn(c) as a submanifold of the sphere Sm(n+2)−1((n + 1)c/(2n)) of constant sectional curvature (n + 1)c/(2n) through an isometric immersion f 1 ◦ ι M , where f 1 is the parallel equivariant minimal
embedding of CPn(c) into Sn(n+2)−1((n + 1)c/(2n)). We here recall the definition and fundamental properties of f 1 . The embedding f 1 is defined by eigenfunctions of the first eigenvalue of the Laplacian ∆ on CP n (c) (for details, see [10, 34]). In submanifold theory, this embedding f 1 is well-known as the only example of a full minimal parallel immersion , i.e., the second fundamental form σ 1 of f 1 is parallel, of a complex projective space endowed with Fubini-Study metric into a Euclidean sphere. The inner product of the first normal space of f 1 is given by (10.1) \σ 1 (X,Y),σ 1 (Z,W)} = −(c/(2n))(X,Y)(Z,W) + (c/4)((X,W)(Y,Z) + ⟨ X, Z ⟩ ⟨ Y, W ⟩ + ⟨ JX, W ⟩ ⟨ JY, Z ⟩ + ⟨ JX, Z ⟩ ⟨ JY, W ⟩ ⟩ for all vectors X, Y, Z, W on CPn (c), where J is the complex structure on CPn (c). Equation (10.1) shows the following properties of f 1 : (i) The embedding f 1 is minimal. (ii) It holds that σ 1 ( JX, JY ) = σ 1 (X, Y ) for all vectors X, Y on CP n (c), i.e., σ 1 is J -invariant. Hence the second fundamental form σ 1 of CP n (c) in S n (n+2)−1 ((n + 1)c/(2n)) is parallel (see Proposition 3). (iii) The length of a normal vector σ 1 ( X, X ) is written as ‖ σ 1 ( X, X )‖ = √(n − 1)c/(2n) for each unit vector X on CPn(c), namely our embedding f 1 is √ [n − 1)c/(2n)-isotropic (cf. [31]). We next explain the embeddings f 1 ◦ ι M : M → S n (n+2)−1 ((n + 1)c/(2n)). This class contains some homogeneous submanifolds of S n (n+2)−1 ((n+1)c/(2n)), that is they are expressed as orbits of some subgroups of the isometry group SO( n ( n + 2)) of the ambient sphere.  In fact, if we take a homogeneous real hypersurface M of CPn(c), the immersion f 1 ◦ ι M gives a homogeneous submanifold M of the sphere.  As a matter of course these homogeneous submanifolds have constant mean curvature in the sphere. On the other hand, the second fundamental form of the immersion f 1 ◦ ι M : M → S n (n+2)−1 ((n + 1)c/(2n)) is not parallel for each real hypersurface M of CP n (c) because CP n (c) admits no real hypersurfaces which are locally symmetric (for example, see [25]). Hence it is natural to pose the following problem: Problem 2. Classify submanifolds ( M ²n−¹, f 1 o t M ) of S n (n+²)−¹((n + 1)c/(2n)) satisfying that the isometric immersion f 1 ◦ ι M has parallel mean curvature vector with respect to the normal connection. The following answer to Problem 2 is a key lemma. Lemma 9 ([23]). Let M²n−1 be a connected real hypersurface of CPn(c),n ≥ 2 through an isometric immersion ι M and f 1 : CPn (c) → Sn(n+2)−1 ((n + 1)c/(2n)) the first standard minimal embedding. Then M is locally congruent to the geodesic sphere G(r) (0 < r < π/√c ) with tan²(√c r/2) = 2n + 1 in CPn(c) if and only if the isometric immersion f 1 ◦ ι M → Sn(n+2)−1((n+1)c/(2n)) has parallel mean curature vector with respect to the normal connection. Moreover, this submanifold (M, f 1 0 ι M ) is homogeneous in the ambient sphere. We find easily the following lemma. Lemma 10 ([23]). The geodesic sphere G(r) of radius r (0 < r < π/√c) with tan²(√c r/2) = 2n + 1 in CPn(c) is a Sasakian manifold with respect to the almost contact metric structure (φ,ξ,η,ց) induced from the Kähler structure (g,J) on CPn(c),n ≥ 2 if and only if c = 8n + 4.  Furthermore, this geodesic sphere is a Sasakian space form of constant φ -sectional curvature 8n + 5. Moreover, we need the following lemma in order to show Statement (2) in The- orem 11. Lemma 11 ([23]). We consider the following isometric embedding f of the geodesic sphere G(r) of radius r (0 < r < π/√√ ) with tan²(√√ r/2) = 2n + 1 in CP n (c) into an N (≥ n (n + 2) − 1)-dimensional sphere S N ( č ) of constant sectional curature c(≤ (n + 1)c/(2n)). (1)  When N > n(n + 2) − 1,  f  is given by ƒ = ι ⊙ (ƒ1 ⊙ ιG(r)) : G(r) ƒ1ºιG(r)  Sm(n+2)−1((n + 1)c/(2n)) → SN(c), where ι is a totally umbilic embedding, so that (n + 1)c/(2n) ≥ č. (2) When N = n(n + 2) − 1, ˜ f is nothing but f 1 ◦ ι G(r), so that (n + 1)c/(2n) = ˜. Then our geodesic sphere is homogeneous in S N ( č ) and it has nonzero parallel mean curvature vector with respect to the normal connection in this sphere. Thus, in view of Lemmas 9, 10 and 11 we establish Theorem 11. At the end of this paper we pose the following open problem. Problem 3. Let f 1 be a minimal parallel full immersion of a complex n -dimensional compact Hermitian symmetric space M n into a Euclidean sphere S 2 n+ p ( č ).  If there exists a real hypersurface ( M 2n−1 , t M ) of M n satisfying that the corresponding submanifold ( M 2 n −1 , f 1 ◦ ι M ) has parallel mean curvature vector with respect to the normal connection in the ambient sphere S 2 n+P ( č ), is our Hermitian symmet- ric space M n holomorphically isometric to a complex projective space CP n (c) of constant holomorphic sectional curvature c = 2ñ= 1) and p = n 2 − 1?
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