Slant Surfaces with Prescribed Gaussian Curvature

Bang-Yen Chen and Luc Vrancken

Abstract A slant immersion was introduced in [2] as an isometric immersion of a Riemannian manifold into an almost Hermitian manifold (M̃, g, J) with constant Wirtinger angle. From J-action point of view, the most natural surfaces in an almost Hermitian manifold are slant surfaces. Flat slant surfaces in complex space forms have been studied in [3, 4]. In this article, we study slant surfaces in complex space forms with arbitrary Gauss curvature. In particular, we prove that, for any θ ∈ (0, \frac{π}{2} ], there exist infinitely many θ-slant surfaces in complex projective plane and in complex hyperbolic plane with prescribed Gaussian curvature.

Mathematics Subject Classifications 2000: 53C40, 53C42, 53B25 Key words: Slant surface, complex space form, slant angle, prescribed mean curvature, prescribed Gaussian curvature.

1 Introduction

Let M be a Riemannian n-manifold and M̃ an almost Hermitian manifold with almost complex structure J and almost Hermitian metric g. We denote by ⟨ , ⟩ the inner product for M as well as for M̃. For any vector X tangent to M, we put

(1.1)  JX = P X + F X,

where P X and F X denote the tangential and normal components of JX, respectively. For each nonzero vector X tangent to M at x, the angle θ(X) between JX and T_{x}M is called the Wirtinger angle of X. The immersion f : M → M̃ is said to be slant if the Wirtinger angle θ(X) is a constant (cf. [2]). The Wirtinger angle θ of a slant immersion is called the slant angle. A slant submanifold with slant angle θ is also called a θ-slant submanifold. Holomorphic and totally real immersions are slant immersions with slant angle 0 and \frac{π}{2} , respectively. A slant immersion is said to be proper slant if it is neither holomorphic nor totally real.

From J-action point of view, slant submanifolds are the simplest and the most natural submanifolds of an almost Hermitian manifold. Slant submanifolds have been studied by many geometers in the last two decades (see, for examples, [1]-[12]). Slant submanifolds arise naturally and play some important roles in the studies of submanifolds of Kählerian manifolds. For example, it was proved in [8] that every surface in a complex space form M̃^{2}(4c) is proper slant if it has constant curvature and nonzero parallel mean curvature vector. Flat slant surfaces in complex space forms have been studied in [3, 4]. There exist ample minimal proper slant surfaces in C^{2} (see, [2]). In contrast, it was proved in [5] that there do not exist minimal proper slant surfaces in complex projective and in complex hyperbolic planes. Furthermore, it was proved in [7] that, for any θ ∈ (0, \frac{π}{2} ] and any given function G = G(x), there exists a θslant surface in the complex Euclidean plane C^{2} with G as the prescribed Gaussian curvature function.

Balkan Journal of Geometry and Its Applications, Vol.7, No.1, 2002, pp. 29-36.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

In this article, we investigate slant surfaces in complex projective and complex hyerbolic planes with arbitrary Gauss curvature. More precisely, we prove the following existence theorem for slant surfaces prescribed Gauss curvature. Theorem 1. Locally, for any given θ ∈ (0, \frac{π}{2} ] and any given function G = G(x), there exist infinitely many θ-slant surfaces in the complex projective plane CP^{2} and in the complex hyperbolic plane CH^{2} with G as the prescribed Gaussian curvature. Remark. The Theorem is false in general if Gaussian curvature were replaced by mean curvature. For example, for any θ ∈ (0, \frac{π}{2} ), there does not exist θ-slant surfaces in CP^{2} and in CH^{2} with zero as the prescribed mean curvature (see [5]).

2 Preliminaries

Let x : M → M̃^{m} be an isometric immersion of a Riemannian n-manifold into a Kählerian m-manifold. Denote by R and R̃ the Riemann curvature tensors of M and M̃^{m}, respectively. Denote by h and A the second fundamental form and the shape operator of the immersion x; and by ∇ and ˜∇ the Levi-Civita connections of M and M̃^{m}, respectively. The second fundamental form h and the shape operator A are related by

(2.1)  ⟨A_{ξ}X, Y ⟩ = ⟨h(X, Y), ξ⟩

for ξ normal to M.

The well-known equation of Gauss is given by

(2.2) R̃(X, Y; Z, W) = R(X, Y; Z, W) + ⟨h(X, Z), h(Y, W)⟩ − ⟨h(X, W), h(Y, Z)⟩

for X, Y, Z, W tangent to M.

For the second fundamental form h, we define its covariant derivative ¯∇h with respect to the connection on T M ⊕ T^{⊥}M by

(2.3)  ( ¯∇_{X}h)(Y, Z) = D_{X}(h(Y, Z)) − h(∇_{X}Y, Z) − h(Y, ∇_{X}Z).

The equation of Codazzi is

(2.4)  (R̃(X, Y)Z)^{⊥} = ( ¯∇_{X}h)(Y, Z) − ( ¯∇_{Y} h)(X, Z),

where (R̃(X, Y)Z)^{⊥} denotes the normal component of R̃(X, Y)Z.

For an endomorphism Q on the tangent bundle of the submanifold, we define its covariant derivative ∇Q by (∇_{X}Q)Y = ∇_{X}(QY) − Q(∇_{X}Y).

For a θ-slant submanifold M in a Kählerian n-manifold M̃^{n}, we have [2]

(2.5)  P^{2} = −(cos^{2} θ)I, ⟨P X, Y ⟩ + ⟨X, P Y ⟩ = 0,

(2.6)  (∇_{X}P)Y = th(X, Y) + A_{F Y} X,

(2.7)  D_{X}(F Y) − F (∇_{X}Y) = fh(X, Y) − h(X, P Y),

where I denotes the identity map and fh(X, Y) is the normal part of Jh(X, Y). If we define a symmetric bilinear T M-valued form α on M by

(2.8)  α(X, Y) = th(X, Y),

then we obtain

(2.9)  h(X, Y) = csc^{2} θ(P α(X, Y) − Jα(X, Y)).

Denote by M̃^{m}(4c) the complete simply-connected Kählerian m-manifold with constant holomorphic sectional curvature 4c. Hence, M̃^{m}(4c) is holomorphically isometric to CP^{m}(4c), C^{m}, or CH^{m}(4c), according to c > 0, c = 0, or c < 0.

For an n-dimensional θ-slant submanifold with θ ≠ 0 in M̃^{n}(4c), the equations of Gauss and Codazzi become

(2.10) \begin{aligned}R(X, Y; Z, W) & = & csc^{2} θ\{⟨α(X, W), α(Y, Z)⟩ − ⟨α(X, Z), α(Y, W)⟩\}+ \\ & + & c\{⟨X, W ⟩ ⟨Y, Z⟩ − ⟨X, Z⟩ ⟨Y, W ⟩ + ⟨P X, W ⟩ ⟨P Y, Z⟩ \\ & − & ⟨P X, Z⟩ ⟨P Y, W ⟩ + 2 ⟨X, P Y ⟩ ⟨P Z, W ⟩\}, \end{aligned}

(2.11)

(∇_{X}α)(Y, Z) + csc^{2} θ{P α(X, α(Y, Z)) + α(X, P α(Y, Z))}+
+(sin^{2} θ)c{⟨X, P Y ⟩ Z + ⟨X, P Z⟩ Y } =
= (∇_{Y} α)(X, Z) + csc^{2} θ{P α(Y, α(X, Z)) + α(Y, P α(X, Z))}+
+(sin^{2} θ)c{⟨Y, P X⟩ Z + ⟨Y, P Z⟩ X}.

We recall the following Existence and Uniqueness Theorems from [7]. Theorem A (Existence Theorem). Let c, θ be two constants with 0 < θ ≤ \frac{π}{2} and M a simply-connected Riemannian n-manifold with inner product ⟨ , ⟩. Suppose there exist an endomorphism P of the tangent bundle T M and a symmetric bilinear T M-valued form α on M such that for X, Y, Z, W ∈ T M, we have

(2.12)  P^{2} = −(cos^{2} θ)I,

(2.13)  ⟨P X, Y ⟩ + ⟨X, P Y ⟩ = 0,

(2.14)  ⟨(∇_{X}P)Y, Z⟩ = ⟨α(X, Y), Z⟩ − ⟨α(X, Z), Y ⟩ ,

(2.15)  \begin{matrix}R(X, Y; Z, W) = csc^{2} θ\{⟨α(X, W), α(Y, Z)⟩ − ⟨α(X, Z), α(Y, W)⟩\}+ \\ +c\{⟨X, W ⟩ ⟨Y, Z⟩ − ⟨X, Z⟩ ⟨Y, W ⟩ + ⟨P X, W ⟩ ⟨P Y, Z⟩ − \\ − ⟨P X, Z⟩ ⟨P Y, W ⟩ + 2 ⟨X, P Y ⟩ ⟨P Z, W ⟩\}, \end{matrix}

and

(2.16)  \begin{aligned}(∇_{X}α)(Y, Z) & + & csc^{2} θ\{P α(X, α(Y, Z)) + α(X, P α(Y, Z))\}+ \\ & + & (sin^{2} θ)c\{⟨X, P Z⟩ Y + ⟨X, P Y ⟩ Z\} \end{aligned}

is totally symmetric. Then there exists a θ-slant isometric immersion from M into M̃^{n}(4c) whose second fundamental form h is given by

(2.17)  h(X, Y) = csc^{2} θ(P α(X, Y) − Jα(X, Y)).

Theorem B (Uniqueness Theorem). Let x^{1}, x^{2} : M → M̃^{n}(4c) be two θ-slant (0 < θ ≤ \frac{π}{2} ) isometric immersions of a connected Riemannian n-manifold M into the complex space form M̃^{n}(4c) with second fundamental form h^{1} and h^{2}. If c ≠ 0 and

(2.18)  ⟨h^{1}(X, Y), Jx_{⋆}^{1}Z⟩ = ⟨h^{2}(X, Y), Jx_{⋆}^{2}Z⟩

for all vector fields X, Y, Z tangent to M, then P_{1} = P_{2} and there exists an isometry ϕ of M̃^{n}(4c) such that x^{1} = ϕ(x^{2}).

3 Proof of Theorem

Let c ∈ \{1, −1\}, θ ∈ (0, \frac{π}{2} ], and let G = G(x) be a differentiable function defined on an open interval I. Consider the following Riccati’s differential equation:

(3.1)  ψ′(x) + ψ^{2}(x) + G(x) = 0.

The well-known existence theorem of first order differential equations implies that equation (3.1) does have many solutions on some neighborhoods of 0.

For each given solution ψ = ψ(x) of (3.1) on an open interval containing 0, we consider the following system of first order differential equations: (3.2)

y′_{1}(x) = {2y_{3}^{2} − 2y_{1}y_{2} + (G(x) − c − 3c cos^{2} θ) sin^{2} θ} csc θ cot θ−
− 3ψy_{1} + 3c sin^{2} θ cos θ,
y′_{2}(x) = {2y_{1}y_{2} − 2y_{3}^{2} − (G(x) − c − 3c cos^{2} θ) sin^{2} θ} csc θ cot θ+
+ (2y_{1} − y_{2})ψ + 3c sin^{2} θ cos θ,
y′_{3}(x) = \frac{ψ}{y_{3}} \{y_{1}^{2} + y_{3}^{2} − y_{1}y_{2} + (G(x) − c − 3c cos^{2} θ) sin^{2} θ\} − 2ψy_{3}+
+ \{\frac{y_{2}}{y_{3}} (y_{1}^{2} + y_{3}^{2} − y_{1}y_{2} + (G(x) − c − 3c cos^{2} θ) sin^{2} θ) − y_{1}y_{3}\} csc θ cot θ

with the initial conditions:

(3.3)  y_{1}(0) = c_{1}, y_{2}(0) = c_{2}, y_{3} = c_{3} ≠ 0.

Although the equations in (3.2) can be written in slightly easier form, however, (3.2) is the most suitable for checking that condition (2.16) will be satisfied.

It is well-known that the system (3.2) with the initial conditions (3.3) has a unique solution y_{1} = μ(x), y_{2} = δ(x), y_{3} = φ(x) on some open interval I containing 0 on which φ(x) is nowhere zero.

We put

(3.4)  λ = \frac{1}{φ}\{μ^{2} + φ^{2} − μδ + (G(x) − c − 3c cos^{2} θ) sin^{2} θ\},

(3.5)  f(x) = exp ( ∫ ^{x} ψ dx),

where ∫ ^{x} ψdx is an anti-derivative of ψ.

Let M be a simply-connected domain containing the origin (0, 0) of the Euclidean plane E^{2}. Assume that the metric on M is the warped product metric:

(3.6)  g = dx ⊗ dx + f^{2}(x)dy ⊗ dy.

Let e_{1} = ∂/∂x, e_{2} = f^{−1}∂/∂y. Then {e_{1}, e_{2}} is an orthonormal frame field of the tangent bundle T M of M. Then, by a direct computation, we have

(3.7)  ∇_{e_{1}}e_{1} = ∇_{e_{1}}e_{2} = 0, ∇_{e_{2}}e_{1} = ψe_{2}, ∇_{e_{2}}e_{2} = −ψe_{1}.

We define a symmetric bilinear T M-valued form α on M by

(3.8)  α(e_{1}, e_{1}) = λe_{1} + μe_{2}, α(e_{1}, e_{2}) = μe_{1} + φe_{2}, α(e_{2}, e_{2}) = φe_{1} + δe_{2}.

Then

(3.9)  ⟨α(X, Y), Z⟩ = ⟨α(X, Z), Y ⟩

for X, Y, Z tangent to M.

It is well-known that the oriented Riemannian 2-manifold (M, g) admits a canonical Kählerian structure J, that is, (g, J) is a Hermitian structure on M with ∇J = 0. If we put P = cos θJ, then we have

(3.10)  P^{2} = −(cos^{2} θ)I, ∇P = 0, ⟨P X, Y ⟩ + ⟨X, P Y ⟩ = 0.

Hence, (M, P, α) satisfies conditions (2.12), (2.13) and (2.14).

On the other hand, by using (3.1)-(3.2), (3.4)-(3.7), and a straightforward long computation, we know that (M, P, α) also satisfies the remaining two conditions stated in the Existence Theorem. Thus, by applying the Existence and Uniqueness Theorem, we know that, up to rigid motions, there exists a unique θ-slant isometric immersion

(3.11)  ϕ_{ψ,c_{1},c_{2},c_{3}} : M → M̃^{2}(4c)

whose second fundamental form h is given by

(3.12)  h(X, Y) = csc^{2} θ(P α(X, Y) − Jα(X, Y)).

From (3.1) and (3.5), we know that the Gaussian curvature of the slant surface is given by G.

Since, for any prescribed Gaussian curvature G = G(x), the function ψ can be chosen to be any of the solutions of the Riccati equation (3.1) and c_{1}, c_{2}, c_{3} to be any three real numbers with c_{3} ≠ 0, we conclude that there exist infinitely many θ-slant surfaces in CP^{2}(4) and in CH^{2}(−4) with G as the prescribed Gaussian curvature.

□

4 Some explicit solutions of the differential system

Here, we provide some explicit solutions of the differential system (3.1)-(3.2) with c = ±1. For simplicity, we assume that G = 0 and we choose ψ = 0 which is the trivial solution of the Riccati equation (3.1). Then the differential system (3.2) reduces to

(4.1)  y′_{1}(x) = 2(y_{3}^{2} − y_{1}y_{2}) csc θ cot θ − c(1 + 3 cos 2θ) cos θ,

(4.2)  y′_{2}(x) = 2(y_{1}y_{2} − y_{3}^{2}) csc θ cot θ + 4c cos θ,

(4.3)  y_{3}y′_{3} = [y_{2}(y_{1}^{2} − y_{1}y_{2} − (c + 3c cos^{2} θ) sin^{2} θ) + (y_{2} − y_{1})y_{3}^{2}] csc θ cot θ.

Summing up (4.1) and (4.2) gives

(4.4)  (y_{1} + y_{2})′(x) = 6c cos θ sin^{2} θ.

Thus, by solving (4.4), we have

(4.5)  y_{2} = 6cx cos θ sin^{2} θ − y_{1} − b

for some constant b. Substituting (4.5) into (4.1) yields

(4.6) y′_{1}(x) = 2(y_{1}^{2} + y_{3}^{2} + by_{1}) csc θ cot θ − 12cxy_{1} cos^{2} θ − c(1 + 3 cos 2θ) cos θ.

If we denote y_{3}^{2} by Φ, then (4.6) becomes

(4.7)  Φ = \frac{y′_{1}}{2} sin θ tan θ − (b + y_{1})y_{1} + \frac{c}{2}(1 + 12xy_{1} cos θ + 3 cos 2θ) sin^{2} θ.

By differentiating (4.6) we find

(4.8) y′′_{1}(x) = 2((b + 2y_{1})y′_{1} + 2y_{3}y′_{3}) csc θ cot θ − 12cy_{1} cos^{2} θ − 12cxy′_{1} cos^{2} θ.

Hence, by substituting (4.3), (4.5) and (4.7) into (4.8), we obtain

(4.9)  y′′_{1}(x) = 4c(2b − 3cx cos θ + 3cx cos 3θ) cot^{2} θ.

After solving the differential equation (4.9), we have

(4.10)  y_{1}(x) = c_{1} + c_{2}x + 4bcx^{2} cot^{2} θ − 8x^{3} cos^{3} θ

for some integration constants c_{1}, c_{2}.

From (4.5) and (4.10) we obtain

(4.11)  y_{2} = 6cx cos θ sin^{2} θ − 4bcx^{2} cot^{2} θ + 8x^{3} cos^{3} θ − b − c_{1} − c_{2}x.

Hence, if we substituting (4.10) and (4.11) into (4.7), we have

(4.12)

y_{3}^{2}(x) = −(c_{1} + c_{2}x + 4bcx^{2} cot^{2} θ − 8x^{3} cos^{3} θ)×
×(b + c_{1} + c_{2}x + 4bcx^{2} cot^{2} θ − 8x^{3} cos^{3} θ)
+\frac{c}{2}[1 + 3 cos 2θ + 12x cos θ(c_{1} + c_{2}x + 4bcx^{2} cot^{2} θ − 8x^{3} cos^{3} θ)] sin^{2} θ
+\frac{1}{2}(c_{2} − 24x^{2} cos^{3} θ + 8bcx cot^{2} θ) sin θ tan θ.

In particular, if c = 1 and b = c_{1} = c_{2} = 0, then we obtain the explicit solution of (3.1)-(3.2):

(4.13)  y_{1} = −8x^{3} cos^{3} θ,

(4.14)  y_{2} = 6x sin^{2} θ cos θ + 8x^{3} cos^{3} θ,

(4.15)  y_{3}^{2} = \frac{1}{2}(1 + 3 cos 2θ − 96x^{4} cos^{4} θ) sin^{2} θ − 3x^{2} sin^{2} 2θ − 64x^{6} cos^{6} θ.

Conversely, it is straightforward to verify that (4.10)-(4.12) satisfies the differential system (4.1)-(4.3).

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Department of Mathematics Michigan State University East Lansing, MI 48824–1027, USA email: bychen@math.msu.edu

Mathematisch Instituut Universiteit Utrecht P.O. Box 80.010 3508 TA UTRECHT, The Netherlands email: vrancken@math.uu.nl