Ruled Surfaces as Pseudospherical Congruences
Two-dimensional ruled surfaces in the spaces of constant curvature Rⁿ, Sⁿ, Hn and in the Riemannian products Sⁿ x R¹, Hⁿ x R¹ are considered. A ruled surface is proved to represent a pseudospherical congruence if and only if it is either an intrinsically flat surface in Sⁿ, or an intrinsically flat...
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irk-123456789-1065482016-10-01T03:01:59Z Ruled Surfaces as Pseudospherical Congruences Gorkavyy, V.O. Nevmerzhitska, O.M. Two-dimensional ruled surfaces in the spaces of constant curvature Rⁿ, Sⁿ, Hn and in the Riemannian products Sⁿ x R¹, Hⁿ x R¹ are considered. A ruled surface is proved to represent a pseudospherical congruence if and only if it is either an intrinsically flat surface in Sⁿ, or an intrinsically flat surface with constant extrinsic curvature in Sⁿ x R¹. 2009 Article Ruled Surfaces as Pseudospherical Congruences / V.O. Gorkavyy,. O.M. Nevmerzhitska // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 359-374. — Бібліогр.: 7 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106548 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Two-dimensional ruled surfaces in the spaces of constant curvature Rⁿ, Sⁿ, Hn and in the Riemannian products Sⁿ x R¹, Hⁿ x R¹ are considered. A ruled surface is proved to represent a pseudospherical congruence if and only if it is either an intrinsically flat surface in Sⁿ, or an intrinsically flat surface with constant extrinsic curvature in Sⁿ x R¹. |
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Gorkavyy, V.O. Nevmerzhitska, O.M. |
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Gorkavyy, V.O. Nevmerzhitska, O.M. Ruled Surfaces as Pseudospherical Congruences Журнал математической физики, анализа, геометрии |
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Gorkavyy, V.O. Nevmerzhitska, O.M. |
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Gorkavyy, V.O. |
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Ruled Surfaces as Pseudospherical Congruences |
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Ruled Surfaces as Pseudospherical Congruences |
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Ruled Surfaces as Pseudospherical Congruences |
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Ruled Surfaces as Pseudospherical Congruences |
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Ruled Surfaces as Pseudospherical Congruences |
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ruled surfaces as pseudospherical congruences |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Ruled Surfaces as Pseudospherical Congruences / V.O. Gorkavyy,. O.M. Nevmerzhitska // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 359-374. — Бібліогр.: 7 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT gorkavyyvo ruledsurfacesaspseudosphericalcongruences AT nevmerzhitskaom ruledsurfacesaspseudosphericalcongruences |
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2025-07-07T18:38:05Z |
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Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 4, pp. 359–374
Ruled Surfaces as Pseudospherical Congruences
V.O. Gorkavyy and O.M. Nevmerzhitska
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:gorkaviy@ilt.kharkov.ua
Received December 22, 2008
Two-dimensional ruled surfaces in the spaces of constant curvature Rn,
Sn, Hn and in the Riemannian products Sn ×R1, Hn ×R1 are considered.
A ruled surface is proved to represent a pseudospherical congruence if and
only if it is either an intrinsically flat surface in Sn, or an intrinsically flat
surface with constant extrinsic curvature in Sn ×R1.
Key words: pseudo-spherical congruence, ruled surface.
Mathematics Subject Classification 2000: 53B25, 53A07.
1. Introduction
The aim of the paper is to discuss two-dimensional ruled surfaces from the
point of view of the theory of pseudospherical congruences.
Let Mn be one of the n-dimensional spaces of constant curvature (Euclidean
space Rn, unite sphere Sn, unite hyperbolic space Hn) or one of the Riemannian
products Sn−1 ×R1, Hn−1 ×R1.
A geodesic congruence in Mn is a diffeomorphism ψ : F 2 → F̃ 2 of two surfaces
in Mn which possesses the following bitangency property: for each point P ∈ F 2
there exists a geodesic of Mn through P and ψ(P ) = P̃ ∈ F̃ 2 which is tangent
to F 2 at P and to F̃ 2 at P̃ .
The geodesic congruence ψ : F 2 → F̃ 2 is said to be pseudospherical if it
satisfies two additional conditions:
(C1) the distance between corresponding points P ∈ F 2 and P̃ ∈ F̃ 2 is equal
to a nonzero constant independent of P , |PP̃ | ≡ l0 6= 0;
(C2) the angle between planes tangent to F 2 and F̃ 2 at corresponding points
is equal to a nonzero constant independent of P , ∠(TP F 2, TP̃ F̃ 2) ≡ ω0 6= 0.
The constants l0 and ω0 are called the parameters of the pseudospherical
congruence ψ.
c© V.O. Gorkavyy and O.M. Nevmerzhitska, 2009
V.O. Gorkavyy and O.M. Nevmerzhitska
This definition corresponds to the classical notion of pseudospherical con-
gruences for n-dimensional submanifolds in (2n − 1)-dimensional spaces of con-
stant curvature [1–3]. Recall that if two n-dimensional submanifolds Φn, Φ̃n in
(2n − 1)-dimensional space of constant curvature are connected by a pseudo-
spherical congruence, then Φn, Φ̃n are of the same constant negative extrinsic
curvature Kext depending on l0, ω0. Moreover, an arbitrary n-dimensional sub-
manifold in (2n−1)-dimensional space of constant curvature admits a large family
of different pseudospherical congruences. These statements, which generalize the
classical results by Backlund, Bianchi, Darboux, were proved by K. Teneblat and
C.-L. Terng [4], Yu.A. Aminov [5]. Using the classical terminology, Φ̃n is called
a Backlund transformation of Φn. This geometric construction was of great im-
portance for the soliton theory, where the development of some fundamental
ideas and principles was initiated. Actually, the n-dimensional submanifolds of
constant negative extrinsic curvature (pseudospherical submanifolds) in (2n−1)-
dimensional spaces of constant curvature represent one of the most illustrative
classical examples of integrable systems [2, 3, 6].
On the other hand, it would be interesting to understand what properties of
pseudospherical congruences still hold if we consider either submanifolds in spaces
of constant curvature without any restrictions on dimension and codimension, or
submanifolds in ambient spaces different from the spaces of constant curvature.
In this paper we consider two-dimensional surfaces in n-dimensional spaces of
constant curvature Rn, Sn, Hn and two-dimensional surfaces in n-dimensional
Riemannian products Sn−1 ×R1, Hn−1 ×R1.
A particular class of surfaces, which is both natural and exceptional from
the point of view of the pseudospherical congruences, consists of ruled surfaces.
Namely, let F 2 ⊂ Mn be an oriented surface ruled by geodesics of Mn. For any
smooth function l on F 2, consider a map ψl : F 2 → F 2 that moves each point
P ∈ F 2 along the corresponding ruling γP ⊂ F 2, which passes through P , to
a point P̃ ∈ F 2 at the distance l from P . Clearly, ψl satisfies the bitangency
condition, so it represents a geodesic congruence. Besides, it will satisfy (C1) if
we set l ≡ l0 > 0, so every point of F 2 moves along a corresponding ruling of F 2
at a fixed distance. Now the problem is to verify the last property (C2). It turns
out that generically (C2) does not hold. However some particular ruled surfaces
are able to represent pseudospherical congruences. Namely, we will demonstrate
some statements which lead to the following conclusions:
1) if F 2 is a ruled surface in one of Rn, Hn, Hn−1 ×R1, then ψl0 : F 2 → F 2
does not represent pseudospherical congruences, for any l0 > 0;
2) if F 2 is a ruled surface in Sn, then ψl0 : F 2 → F 2 represents a pseudo-
spherical congruence if and only if F 2 is an intrinsically flat surface;
3) if F 2 is a ruled surface in Sn−1 × R1, then ψl0 : F 2 → F 2 represents
a pseudospherical congruence if and only if F 2 is an intrinsically flat surface with
360 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
constant negative extrinsic curvature.
Notice that there exist intrinsically flat ruled surfaces in Sn−1×R1 with non-
constant extrinsic curvature. (It would be interesting to verify that there exist
ruled surfaces in Sn−1 × R1 with constant extrinsic curvature and nonconstant
intrinsic curvature.) Thus a surface in Sn−1 × R1 must be referred to as pseu-
dospherical if it has constant intrinsic curvature and constant negative extrinsic
curvature. We hope that one can construct a consistent theory of pseudospherical
congruences and Backlund transformations for two-dimensional pseudospherical
surfaces in Sn−1 ×R1 and Hn−1 ×R1.
We emphasize that the constant ω0 in (C2) is subject to the constrain
0 < ω0 ≤ π
2 . On the other hand, if we allow ω0 to vanish, then ruled surfaces with
vanishing extrinsic curvature would appear as pseudospherical congruences too:
for any of these ruled surfaces its tangent plane is stationary along any ruling.
The paper is organized as follows. In Section 2 we consider a rather trivial
case of ruled surfaces in Rn. Sections 3–5 are devoted to the most interesting
case of ruled surfaces in Sn−1 ×R1 and Sn. In Section 6. we analyze briefly the
ruled surfaces in Hn−1 × R1 and Hn. All considerations are local, all functions
are supposed to be sufficiently smooth.
2. Ruled Surfaces in Rn
Let F 2 be an oriented ruled surface in n-dimensional Euclidean space Rn,
n ≥ 3. This surface may be represented by the position vector
r(u, v) = r0(v) + ua(v),
where r0(v) is a vector function representing a base curve of F 2, whereas a(v) is
a unit vector function determining the directions of straight lines (rulings) which
sweep out the surface F 2. Without loss of generality, one can assume that the
base curve u = 0 is orthogonal to the rulings v = const of F 2,
〈r′0, a〉 = 0.
Besides, we will assume that v is the arc length of the base curve u = 0, so |r′0| ≡ 1.
Due to the described particular choice of local coordinates u, v, the induced metric
on F 2 is ds2 = du2 + g22dv2, so u, v form a semigeodesic coordinate system in
F 2, and u is an arc length for every ruling v = const.
Proposition 1. Let F 2 ⊂ Rn be a ruled surface represented by a position-
vector r(u, v) = r0(v) + ua(v), which satisfies |a| = |r′0| ≡ 1 and 〈r′0, a〉 = 0.
The surface F 2 has a constant Gauss curvature if and only if it is intrinsically
flat, K ≡ 0. The ruled surface F 2 is intrinsically flat if and only if [r′0, a] = 0.
The proof of this statement is a rather simple task because of the simple form
of ds2 described above.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 361
V.O. Gorkavyy and O.M. Nevmerzhitska
Now fix a positive constant l0 and consider a regular map Φl0 : F 2 → F 2,
which moves every point of F 2 along the corresponding ruling of F 2 at the dis-
tance l0. This map Φl0 is represented by the following formula:
r̃ = r + l0a = r0 + (u + l0)a. (1)
Evidently, Φl0 satisfies both the bitangency condition and (C1). Let us
analyze the condition(C2): for this purpose choose an arbitrary point P ∈ F 2
and calculate the angle ω between the planes tangent to F 2 at points P and
P̃ = Φl0(P ).
Recall that if two-dimensional planes R2
1, R2
2 in Rn have a common straight
line R1
12 = R2
1 ∩ R2
2, then the angle between R2
1 and R2
2 is defined as the angle
between straight lines R1
1 ⊂ R2
1 and R1
2 ⊂ R2
2 orthogonal to R1
12.
The tangent plane TP F 2 at P is spanned by the vectors
ru = (r0(v) + ua(v))′u = a, (2)
rv = r′0 + ua′. (3)
The tangent plane TP̃ F 2 at P̃ is spanned by the vectors
r̃u = ru + (l0a)u = a, (4)
r̃v = rv + (l0a)v = r′0 + ua′ + l0a
′ = r′0 + a′(u + l0). (5)
Remark that a is orthogonal to rv and r̃v.
The tangent planes TP F 2 and TP̃ F 2 have a common straight line, which is
just the corresponding ruling γP ⊂ F 2. Clearly, γP ⊂ F 2 is spanned by a,
therefore the angle ω between TP F 2 and TP̃ F 2 is determined by the vectors rv
and r̃v orthogonal to a. So ω is equal to a constant ω0 if and only if
〈rv, r̃v〉2 = |rv|2 |r̃v|2 cos2 ω0. (6)
By (3) and (5), equality (6) may be rewritten as follows:
(1 + A(2u + l0) + u(u + l0)B)2
= (1 + 2uA + u2B)(1 + 2Au + 2l0A + Bu2 + 2Bul0 + Bl20) cos2 ω0,
(7)
where A(v) = 〈r′0, a′〉, B(v) = 〈a′, a′〉. This equality, which is polynomial with
respect to u, holds if the coefficients at corresponding degrees of u coincide. Hence
we obtain five equalities:
1 + 2Al0 + A2l20 = (1 + 2Al0 + Bl20) cos2 ω0, (8)
4A + 4A2l0 + 2Bl0 + 2ABl20 = (4A + 4A2l0 + 2Bl0 + 2ABl20) cos2 ω0, (9)
4A2 + 2B + 6ABl0 + B2l20 = (4A2 + 2B + 6ABl0 + B2l20) cos2 ω0, (10)
4AB + 2B2l0 = (4AB + 2B2l0) cos2 ω0, (11)
B2 = B2 cos2 ω0. (12)
362 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
Since ω0 ∈ (0, π
2 ], equality (12) holds if and only if B = 0. In this case equality
(10) reads 4A2 = 4A2 cos2 ω0, therefore A = 0. Then (8) becomes 1 = cos2 ω0,
so it contradicts to the assumption ω0 ∈ (0, π
2 ]. Therefore (6) does not hold if
ω0 ∈ (0, π
2 ].
Theorem 1. Let F 2 be a ruled surface in Rn. Let Φl0 : F 2 → F 2 be a regular
map that moves every point P ∈ F 2 along a corresponding ruling of F 2 at a fixed
constant distance l0 > 0. Then Φl0 does not satisfy (C2) for any ω0 ∈ (0, π
2 ].
Thus the ruled surfaces in Rn do not represent pseudospherical congruences.
On the other hand, if one allows ω0 to be equal to 0, then (C2) turns out to
be less restrictive. Namely, it is well known that an arbitrary ruled surface with
vanishing Gauss curvature in Rn is tangentially degenerate, its tangent plane is
stationary along any ruling.
3. Ruled Surfaces in Sn ×R1
Now let us discuss ruled surfaces in the Riemannian product Sn×R1, where Sn
denotes the unit sphere. We will view Sn×R1 as the hypersurface in Rn+1⊕R1 =
Rn+2 represented by the equation (x1)2 + . . .+(xn+1)2 = 1 in terms of Cartesian
coordinates (x1, . . . , xn+2) in Rn+2. The unit sphere Sn will be considered as the
section of Sn×R1 by the horizontal hyperplane Rn+1 ⊂ Rn+2 given by xn+2 = 0.
Every curve {P} × R1 ⊂ Sn × R1, where P is a point of Sn, is a geodesic
curve of Sn × R1 usually referred to as a vertical one. It may be represented
by the position vector r(u) = ρ + ue, here e = (0, . . . , 0, 1), whereas ρ stands
for the position vector of P . A ruled surface in Sn × R1 swept out by vertical
geodesics is a cylindrical surface, a vertical cylinder, which may be represented
in the following form:
r(u, v) = ρ(v) + ue, (13)
here ρ(v) determines the base curve of the cylinder. Evidently, the cylindrical
surfaces in question are intrinsically and extrinsically flat, Kint = Kext ≡ 0.
Moreover, when a point on such a cylindrical surface moves along the correspon-
ding ruling, the tangent plane is stationary.
An arbitrary nonvertical geodesic curve γ in Sn ×R1 may be represented by
the position vector r(u) = a cosu+ b sinu+e(pu+q), where a, b are orthonormal
vectors in the horizontal hyperplane Rn+1, p and q are some constants. Notice
that the position vector r∗(u) = a cosu+b sinu represents a great circle γ∗ in the
unit sphere Sn ⊂ Rn+1. Evidently, γ∗ is obtained as the intersection of Sn with
a two-plane spanned by a and b. It is natural to say that γ∗ is the base of γ.
An arbitrary ruled surface F 2 in Sn ×R1 swept out by nonvertical geodesics
may be represented in the following way:
r(u, v) = a(v) cosu + b(v) sinu + e(p(v)u + q(v)), (14)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 363
V.O. Gorkavyy and O.M. Nevmerzhitska
where a(v), b(v) are orthonormal vector functions, p(v) and q(v) are some func-
tions which depend on v only. We will assume that all functions in question are
sufficiently smooth. Since a(v), b(v) are orthonormal, we have
|a| = 1, |b| = 1, 〈a, b〉 = 0. (15)
Moreover, without loss of generality, one can specify the choice of a(v), b(v) in
such a particular way that the following additional conditions will be satisfied:
〈a′, b〉 = 0, 〈a, b′〉 = 0. (16)
Namely, if it is necessary, one can replace a, b by a] = a cos ζ + b sin ζ, b] =
−a sin ζ + b cos ζ, where ζ(v) is some suitably chosen function. We will always
assume that (15)–(16) hold. Let us emphasize that the rulings of F 2 are given
by v = const.
R e m a r k. If a′ = b′ = 0, i.e., if a and b are constant, then F 2 is swept
out by nonvertical geodesics v = const as well as by vertical geodesics u = const.
Hence F 2 is a vertical cylinder, its base curve is a great circle in Sn, but its
parametrization (14) is different from (13).
Now let us classify the ruled surfaces in Sn×R1 with constant intrinsic (Gauss)
curvature.
Proposition 2. A regular ruled surface in Sn×R1 with position vector (14)
that satisfies (15)–(16) has a constant Gauss curvature if and only if:
1) |a′| = |b′|, 〈a′, b′〉 = 0, and p is constant, p ≡ p0,
or
2) [a′, b′] = 0, and p, q are constant, p ≡ p0, q ≡ q0,
or
3) a′ = b′ = 0.
Moreover, Kint = 0 in the cases 1) and 3), and Kint = 1
1+p2
0
in the case 2).
P r o o f. The metric of F 2 reads
ds2 = (1+p2)du2+2p(p′u+q′)dudv+
(
X cos 2u + Y sin 2u + Z + (p′u + q′)2
)
dv2,
(17)
where X = |a′|2−|b′|2
2 , Y = 〈a′, b′〉, Z = |a′|2+|b′|2
2 .
The intrinsic curvature Kint of F 2 may be expressed in terms of coordinates
u, v and functions X, Y , Z, p, q depending on v. The expression for Kint is
rather cumbersome, one can suggest to apply some standard computer programs
of symbolic calculus to derive the expression in question. Anyway, it is easy
to verify that the intrinsic curvature is constant, Kint ≡ K0, if and only if an
equality of the following form holds:
P0 + P1 cos 2u + P2 sin 2u + P3 cos 4u + P4 sin 4u = 0, (18)
364 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
where Pi are some functions, which are polynomial with respect to u. Clearly,
(18) is equivalent to the system of five equalities P0 = 0, . . . , P4 = 0. In particular,
we have
P3 = −1
2
(X2 − Y 2)(1 + p2)
(
K0(1 + p2)− 1
)
= 0, (19)
P4 = −2XY (1 + p2)
(
K0(1 + p2)− 1
)
= 0. (20)
It follows from (19) that either X = 0, Y = 0, or K0(1 + p2)− 1 = 0.
In the first case, when X = Y = 0, the equalities P1 = 0, P2 = 0 hold too,
whereas P0 = 0 reads
(p′)2Z + K0
(
(p′)2 · u2 + 2p′q′ · u + (q′)2 + (1 + p)2Z
)2 = 0. (21)
This is a polynomial equality of the 4th order with respect to u, it holds identical-
ly if and only if all its coefficients vanish. It is easy to verify that (21) holds if and
only if either p′ ≡ 0 and K0 = 0, or Z ≡ 0 and K0 = 0. Recall the expressions
for X, Y to see that X = 0, Y = 0 are equivalent to |a′|2 = |b′|2, 〈a′, b′〉 = 0,
whereas X = 0, Y = 0, Z = 0 are equivalent to a′ = b′ ≡ 0.
In the second case, when K0(1+p2)−1 = 0, the function p has to be constant,
p ≡ p0, and K0 = 1
1+p2
0
∈ (0, 1]. Similarly to the first case, the equalities P1 = 0,
P2 = 0 hold too, whereas P0 = 0 reads
Z2 −X2 − Y 2 +
(
1
1 + p2
0
)2
(q′)4 + 2
1
1 + p2
0
Z(q′)2 = 0. (22)
Since Z = |a′|2+|b′|2
2 ≥ 0 and Z2 −X2 − Y 2 = |[a′, b′]|2 ≥ 0, it is easy to see that
(22) holds if and only if q′ ≡ 0, [a′, b′] ≡ 0, q.e.d.
Thus there are three different classes of ruled surfaces with constant intrinsic
curvature in Sn ×R1:
1) intrinsically flat ruled surfaces represented by (14) with a, b, p, q satisfying
(15)–(16) and |a′| = |b′| 6= 0, 〈a′, b′〉 = 0, p ≡ p0;
2) ruled surfaces represented by (14) with a, b, p, q satisfying (15)–(16) and
[a′, b′] = 0, p ≡ p0, q ≡ q0;
3) intrinsically flat vertical cylinders represented by (13) or by (14) with a, b
satisfying a′ = b′ = 0.
Now let us analyze what kind of ruled surfaces with constant intrinsic curva-
ture in Sn ×R1 are of constant extrinsic curvature.
Proposition 3. Let F 2 ⊂ Sn×R1 be a regular intrinsically flat ruled surface
with position vector (14) that satisfies (15)–(16) and |a′| = |b′| 6= 0, 〈a′, b′〉 = 0,
p ≡ p0. Then the extrinsic curvature of F 2 is equal to
Kext = − |a′|2
(1 + p2
0)|a′|2 + (q′)2
.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 365
V.O. Gorkavyy and O.M. Nevmerzhitska
P r o o f. The plane tangent to F 2 is spanned by the vectors
∂ur = −a sinu + b cosu + ep0,
∂vr = a′ cosu + b′ sinu + eq′,
since p ≡ p0 by assumption. The vector
N0 = a cosu + b sinu
is normal to Sn ×R1 ⊂ Rn+2.
By the above assumptions, the metric of F 2 reads
ds2 = (1 + p2
0)du2 + 2p0q
′dudv + (|a′|2 + (q′)2)dv2.
Let II = L11du2 + 2L12dudv + L22dv2 denote the normal-valued second fun-
damental form of F 2 ⊂ SN ×R1. The second partial derivative
∂uur = −a cosu− b sinu
is evidently equal to −N0, so L11 = 0. On the other hand, the second partial
derivative
∂uvr = −a′ sinu + b′ cosu
is orthogonal to ∂ur, ∂vr and N0. Therefore L12 = ∂uvr, so the extrinsic curvature
of F 2 is the following:
Kext =
−|L12|2
g11g22 − (g12)2
=
−|a′|2
(1 + p2
0)|a′|2 + (q′)2
.
Thus we see that generically the extrinsic curvature of the intrinsically flat
ruled surfaces in question is not constant, the constancy of Kext results in some
additional restrictions.
Corollary. Let F 2 ⊂ Sn×R1 be as in Proposition 3. The extrinsic curvature
of F 2 is constant if and only if q′ = c|a′|. In this case Kext = − 1
1 + p2
0 + c2
.
E x a m p l e. Consider a ruled surface F 2 in S3 × R1 ⊂ R5 represented by
the following position vector:
r(u, v) =
cos v
sin v
0
0
0
cosu +
0
0
cos v
sin v
0
sinu +
0
0
0
0
p0u + q(v)
.
366 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
Its intrinsic curvature is Kint = 0, whereas the extrinsic curvature Kext is equal
to − 1
1+p2
0+(q′)2 , so Kext is constant if and only if q(v) is a linear function. Besides,
for an arbitrary K̂0 ∈ [−1, 0) one can choose appropriate p0 and q(v) such that
Kext ≡ K̂0.
As for other classes of ruled surfaces with constant intrinsic curvature in
Sn ×R1, the situation is more trivial.
Proposition 4. Let F 2 ⊂ Sn×R1 be a regular intrinsically flat ruled surface
with position vector (14) that satisfies (15)-(16) and [a′, b′] = 0, p ≡ p0, q ≡ q0.
Then the extrinsic curvature of F 2 vanishes, Kext = 0.
P r o o f. The plane tangent to F 2 is now spanned by the vectors
∂ur = −a sinu + b cosu + ep0,
∂vr = a′ cosu + b′ sinu.
The vector N0 = a cosu + b sinu is normal to Sn ×R1 ⊂ Rn+2.
Let II = L11du2 + 2L12dudv + L22dv2 still denote the normal-valued second
fundamental form of F 2 ⊂ SN ×R1. The second partial derivative
∂uur = −a cosu− b sinu
is equal to −N0, so L11 = 0. Besides, the second partial derivative
∂uvr = −a′ sinu + b′ cosu
is collinear to ∂vr, since [a′, b′] = 0. Therefore L12 = 0. Hence Kext ≡ 0, q.e.d.
Finally, it is evident that the vertical cylinders in Sn×R1 are both intrinsically
and extrinsically flat, Kint = Kext ≡ 0.
4. Pseudospherical Congruences in Sn ×R1
Now let us analyze when a ruled surface in Sn×R1 represents a pseudosphe-
rical congruence. So, let F 2 ⊂ Sn × R1 be a ruled surface represented as in the
previous section by the position vector (14) that satisfies (15)–(16). Fix a constant
l0 and consider the map ψl0 : F 2 → F 2 which moves each point P ∈ F 2 along the
corresponding ruling γP ⊂ F 2, which passes through P , to a point P̃ ∈ F 2 at the
distance l0 from P . Take into account (14) to see that ψl0 may be represented as
follows:
r(u, v) = a(v) cos(u + α) + b(v) sin(u + α) + e (p(v)(u + α) + q(v)) , (23)
where α = l0√
1+p2
depends on v.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 367
V.O. Gorkavyy and O.M. Nevmerzhitska
Clearly, ψl0 satisfies the bitangency condition and represents a geodesic con-
gruence. Besides, it satisfies the requirement (C1). The question is to verify the
condition (C2): one has to check when the angle between TP F 2 and TP̃ F 2 is
constant, ∠(TP F 2, TP̃ F 2) ≡ ω0 ∈ (0, π
2 ].
To be more precise, let us consider the map Π : TP Sn × R1 → TP̃ Sn × R1
generated by the parallel translation in Sn×R1 from P to P̃ along γP . Evidently,
the vector γ̇P (P ) tangent to the geodesic γP at P is mapped by Π to the vector
γ̇P (P̃ ) tangent to γP at P̃ . The plane TP F 2 tangent to F 2 at P is mapped by
Π to a two-plane V ⊂ TP̃ Sn × R1. Since γ̇P (P ) ∈ TP F 2 and γ̇P (P̃ ) ∈ TP̃ F 2,
the intersection of two-planes V and TP̃ F 2 is just the straight line in TP̃ Sn×R1
spanned by γ̇P (P̃ ).
The angle ∠(TP F 2, TP̃ F 2) is defined as the angle between V and TP̃ F 2.
In order to calculate the latter, one has to find vectors Ẑ ∈ V and Z̃ ∈ TP̃ F 2
orthogonal to γ̇P (P̃ ). In particular, ∠(V, TP̃ F 2) ≡ ω0 if and only if the following
equality holds at every point P :
〈Ẑ, Z̃〉2 = cos2 ω0|Ẑ|2|Z̃|2. (24)
Let us find Ẑ and Z̃ and then analyze (24).
The tangent plane TP F 2 is spanned by the vectors
∂ur = −a sinu + b cosu + ep, (25)
∂vr = a′ cosu + b′ sinu + e
(
p′u + q′
)
. (26)
Clearly, ∂ur is equal to γ̇P (P ). Consider the vector Z = −〈∂ur, ∂vr〉∂ur+|∂ur|2∂vr
in TP F 2 which is orthogonal to ∂ur. By (25), we have
Z = −p(p′u + q′) (−a sinu + b cosu)
+ (1 + p2)
(
a′ cosu + b′ sinu
)
+ e(p′u + q′).
Since Z ∈ TP F 2 is orthogonal to γ̇P (P ), then the image of Z under Π belongs
to V and it is orthogonal to γ̇P (P̃ ). Hence one can set Ẑ = Π(Z). The parallel
translation Π along γP in Sn ×R1 is generated by a vertical translation in Rn+2
followed by an orthogonal transformation in Rn+2 which acts as a rotation in the
two-plane spanned by the vectors a, b and as the identity map in the orthogonal
compliment of this two-plane. Therefore we obtain
Ẑ = −p(p′u + q′) (−a sin(u + α) + b cos(u + α))
+(1 + p2) (a′ cosu + b′ sinu) + e(p′u + q′),
(27)
since a′, b′ and e are orthogonal to a and b.
368 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
On the other hand, the tangent plane TP̃ F 2 is spanned by the vectors
∂ur̃ = −a sin(u + α) + b cos(u + α) + ep,
∂v r̃ = a′ cos(u + α) + b′ sin(u + α) + e (p′(u + α) + q′)
+α′ (−a sin(u + α) + b cos(u + α) + ep) .
(28)
Evidently, ∂ur̃ is equal to γ̇P (P̃ ). The vector Z̃ = −〈∂ur̃, ∂v r̃〉∂ur̃ + |∂ur̃|2∂v r̃ in
TP̃ F 2 is orthogonal to ∂ur̃. By (28), we have
Z̃ = −p(p′(u + α) + q′) (−a sin(u + α) + b cos(u + α))
+(1 + p2) (a′ cos(u + α) + b′ sin(u + α)) + e(p′(u + α) + q′).
(29)
Substituting (27) and (29) into (24), we get a cumbersome equality of the
following form:
Q0 + Q1 cos 2u + Q2 sin 2u + Q3 cos 4u + Q4 sin 4u = 0, (30)
where Qi are polynomial with respect to u with coefficients expressed in terms
of X(v) = |a′|2−|b′|2
2 , Y (v) = 〈a′, b′〉, Z(v) = |a′|2+|b′|2
2 , p(v), q(v) and α(v). Since
Qi are polynomial in u, equality (30) is equivalent to the system of five equalities
Q0 = 0, . . . , Q4 = 0. In particular, we have
Q3 = −1
2 sin2 ω0(1 + p2)4
(
(Y 2 −X2) cos(2α)− 2XY sin(2α)
)
= 0,
Q4 = 1
2 sin2 ω0(1 + p2)4
(
(Y 2 −X2) sin(2α) + 2XY cos(2α)
)
= 0.
(31)
Since ω0 ∈ (0, π
2 ] by assumption, (31) is reduced to a linear system with
respect to Y 2 −X2 and 2XY . The only solution is Y 2 −X2 = 0, 2XY = 0, i.e.,
X = 0, Y = 0. In this case the equalities Q1 = 0, Q2 = 0 hold too, whereas
Q0 = 0 reads as follows:
Q04u
4 + Q03u
3 + Q02u
2 + Q01u + Q00 = 0, (32)
where the coefficients Q0i are expressed in terms of Z(v), p(v), q(v) and α(v).
In particular, Q04 = −2 sin2 ω0(p′)4, so if (32) holds, then p ≡ p0. On the other
hand, if p ≡ p0, then (32) becomes as follows:
(
(1 + p2
0)Z cosα + (q′)2
)2 = cos2 ω0
(
(1 + p2
0)Z + (q′)2
)2
. (33)
Thus the angle between TP F 2 and TP̃ F 2 is constant and equal to ω0 ∈ (0, π
2 ]
if and only if X ≡ 0, Y ≡ 0, p ≡ p0 and (33) holds. In other words, it means that
|a′| = |b′|, 〈a′, b′〉 ≡ 0, p ≡ p0 hold together with
(
(1 + p2
0)|a′|2 cosα + (q′)2
)2 = cos2 ω0
(
(1 + p2
0)|a′|2 + (q′)2
)2
, (34)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 369
V.O. Gorkavyy and O.M. Nevmerzhitska
since Z = 1
2(|a′|2 + |b′|2) = |a′|2. Due to Proposition 2, |a′| = |b′|, 〈a′, b′〉 ≡ 0,
p ≡ p0 mean exactly that F 2 is intrinsically flat. Moreover, substituting q′ =
c|a′| into (34), we obtain
(
(1 + p2
0) cos α + c2
)2 = cos2 ω0
(
1 + p2
0 + c2
)2, so c is
constant. Therefore the extrinsic curvature of F 2 is constant, Kext = − 1
1+p2
0+c2
,
by Proposition 3.
Thus we proved the following statement.
Theorem 2. The map ψl0 : F 2 → F 2 represents a pseudospherical con-
gruence if and only if F 2 is an intrinsically flat surface with constant extrinsic
curvature.
We should emphasize that if an intrinsically flat surface with constant extrin-
sic curvature in Sn×R1 is given, then for any l0 6= 2πk
√
1 + p2
0 one can construct
a well-defined pseudospherical congruence ψl0 : F 2 → F 2, whose parameters l0,
ω0 necessarily satisfy the equality
(
(1 + p2
0) cos
l0√
1 + p2
0
+ c2
)2
= cos2 ω0
(
1 + p2
0 + c2
)2
.
E x a m p l e. Consider a ruled surface F 2 in S3 × R1 ⊂ R5 represented by
the following position vector:
r(u, v) =
cos v
sin v
0
0
0
cosu +
0
0
cos v
sin v
0
sinu +
0
0
0
0
p0u + q0v + q1
.
Its intrinsic curvature is Kint ≡ 0, whereas the extrinsic curvature Kext is equal
to − 1
1+p2
0+q2
0
. An arbitrary l0 > 0 given, the map ψl0 : F 2 → F 2 represents
a pseudospherical congruence with
cos2 ω0 =
(
1− 1 + p2
0
1 + p2
0 + q2
0
(1− cos
l0√
1 + p2
0
)
)2
.
5. Ruled Surfaces and Pseudospherical Congruences in Sn
Since the unit sphere Sn may be viewed as a horizontal section of Sn × R1,
the ruled surfaces in Sn are just the ruled surfaces in Sn × R1 with p ≡ 0 and
q ≡ q0. This gives us a simple way to adopt the ideas, methods and results
obtained above in Sections 3–4.
370 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
An arbitrary ruled surface F 2 in Sn swept out by great circles of Sn may be
represented in the following way:
r(u, v) = a(v) cos u + b(v) sin u, (35)
where a(v), b(v) are orthonormal vector functions
|a| = 1, |b| = 1, 〈a, b〉 = 0. (36)
Moreover, without loss of generality, one can specify the choice of a(v), b(v) in
such a particular way that the following additional conditions will be satisfied:
〈a′, b〉 = 0, 〈a, b′〉 = 0. (37)
The metric of F 2 reads ds2 = du2 +(a′ cosu+ b′ sinu)2dv2, so (u, v) is a semi-
geodesic local coordinate system in F 2. The coordinate curves v = const are
rulings of F 2, and u is the arc length for any of these rulings.
The following statement describes the ruled surfaces with constant intrinsic
curvature in Sn. This is Proposition 2 rewritten for the particular case of p ≡ 0,
q ≡ q0.
Proposition 2[. A regular ruled surface F 2 in Sn with position vector (35)
that satisfies (36)–(37) has a constant Gauss curvature if and only if either:
1) |a′| = |b′|, 〈a′, b′〉 = 0,
or
2) [a′, b′] = 0.
Moreover Kint ≡ 0 in the Case 1), and Kint ≡ 1 in the Case 2).
Clearly, for an arbitrary surface in the unit sphere Sn one has Kint = Kext+ 1,
so the constancy of Kint is equivalent to the constancy of Kext. Therefore
an intrinsically flat surface in Sn is pseudospherical, its extrinsic curvature is con-
stant negative, Kext ≡ −1. On the other hand, if Kint ≡ 1, then Kext ≡ 0. Let us
mention two trivial examples: the Clifford tori are intrinsically flat (Kint ≡ 0),
whereas the totally geodesic two-dimensional spheres have the vanishing extrinsic
curvature, so Kint ≡ 1. A rather substantial discussion of surfaces with constant
curvature in Sn one can find, for example, in [7].
Now, fix a constant l0 and consider the map ψl0 : F 2 → F 2, which moves each
point P ∈ F 2 along the corresponding ruling of F 2 at the distance l0. This map
may be represented as follows:
r̃(u, v) = a(v) cos(u + l0) + b(v) sin(u + l0).
Evidently, ψl0 satisfies the bitangency condition as well as the requirement (C1).
The question is when ψl0 satisfies the requirement (C2) and represents a pseudo-
spherical congruence. The answer is given in Theorem 2, where we have to set
p ≡ 0, q ≡ q0.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 371
V.O. Gorkavyy and O.M. Nevmerzhitska
Theorem 2[. The map ψl0 : F 2 → F 2 represents a pseudospherical congru-
ence if and only if F 2 is intrinsically flat, Kint ≡ 0. Besides, the parameters ω0
and l0 of ψl0 are related by the equality cos2 ω0 = cos2 l0.
6. Ruled Surfaces and Pseudospherical Congruences
in Hn and Hn ×R1
Similarly to the spherical case one can consider ruled surfaces and pseudo-
spherical congruences in the hyperbolic space Hn and in the Riemannian product
Hn × R1. The hyperbolic space Hn is viewed as the hypersurface in Minkowski
space Mn+1 represented by the relations |x|2 = −(x1)2 + . . . + (xn+1)2 = −1,
x1 > 0 in terms of Cartesian coordinates (x1, . . . , xn+1) in Mn+1. The mani-
fold Hn × R1 is viewed as the hypersurface in Mn+2 = Mn+1 ⊕ R1 represented
by the same relations in terms of Cartesian coordinates (x1, . . . , xn+1, xn+2) in
Mn+2. Clearly, Hn may be interpreted as the section of Hn×R1 by an arbitrary
horizontal hyperplane Mn+1 ⊂ Mn+2 given by xn+2 = const.
A vertical cylinder in Hn × R1, which is swept out by vertical geodesics of
Hn ×R1, is represented in the following way:
r(u, v) = ρ(v) + ue; (38)
here ρ(v) determines the base curve of cylinder, it belongs to Hn = Hn × {0},
and e = (0, . . . , 0, 1). It is evident that the vertical cylinders are intrinsically and
extrinsically flat, Kint = Kext ≡ 0. Moreover, when a point on such a cylindrical
surface moves along the corresponding ruling, the tangent plane is stationary.
An arbitrary ruled surface F 2 in Hn×R1 swept out by nonvertical geodesics
may be represented in the following way:
r(u, v) = a(v) coshu + b(v) sinhu + e(p(v)u + q(v)), (39)
where a(v), b(v) are orthonormal vector functions with values in Mn+1 =
Mn+1 × {0} which satisfy
|a|2 = −1, |b|2 = 1, 〈a, b〉 = 0, (40)
p(v) and q(v) are some arbitrary functions depending on v. Without loss of
generality, one can always specify the choice of a(v), b(v) in such a way that the
following additional conditions will be satisfied:
〈a′, b〉 = 0, 〈a, b′〉 = 0. (41)
Notice that a is timelike, whereas b is spacelike. Since a′ and b′ are orthogonal
to a, they are spacelike or vanishing.
372 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Ruled Surfaces as Pseudospherical Congruences
Proposition 5. A regular ruled surface in Hn×R1 with position vector (39),
which satisfies (40)–(41), is intrinsically flat if and only if a′ = b′ = 0.
P r o o f. The metric form of F 2 reads
ds2 = (1 + p2)du2 + 2p(p′u + q′)dudv+
+
(
X cosh 2u + Y sinh 2u + Z + (p′u + q′)2
)
dv2,
where X = |a′|2+|b′|2
2 , Y = 〈a′, b′〉, Z = |a′|2−|b′|2
2 .
It is easy to verify that F 2 is intrinsically flat, Kint ≡ 0, if and only
if an equality of the following form holds:
P0 + P1 cosh 2u + P2 sinh 2u + P3 cosh 4u + P4 sinh 4u = 0, (42)
here Pi are some functions which are polynomial with respect to u. The last
equality is equivalent to the system of five equalities P0 = 0, . . . , P4 = 0. In par-
ticular, we have
P3 = −1
2
(X2 + Y 2)(1 + p2) = 0, P4 = −2XY (1 + p2) = 0,
so X = 0 and Y = 0. Recall that X = |a′|2+|b′|2
2 , Y = 〈a′, b′〉. Since a′ and b′ are
either spacelike or vanishing, then X = Y = 0 if and only if a′ = b′ = 0.
On the other hand, if a′ = 0 and b′ = 0, then (42) holds without any additional
requirements, so Kint ≡ 0, q.e.d.
Notice that if a′ = b′ = 0, i.e., if a and b are constant, then F 2 is swept out by
nonvertical geodesics v = const as well as by vertical geodesics u = const. Hence
F 2 is a vertical cylinder, its base curve is a geodesic of Hn, but its parametrization
(38) is different from (39).
Thus only intrinsically flat ruled surfaces in Hn × R1 are vertical cylinders.
Since Hn may be viewed as a horizontal section of Hn × R1, we obtain that
there are no intrinsically flat ruled surfaces in Hn. It seems to be true that only
ruled surfaces of constant intrinsic curvature in Hn are totally geodesic ones.
An interesting problem is to classify the ruled surfaces with constant intrinsic (or
extrinsic) curvature in Hn ×R1.
Let us briefly discuss when a ruled surface in Hn × R1 (or in Hn) repre-
sents a pseudospherical congruence. A ruled surface F 2 in Hn × R1 (or in Hn)
given, let ψl0 : F 2 → F 2 be a regular map which moves each point of F 2 along
the corresponding ruling of F 2 at a constant distance l0. Clearly, it satisfies
the bitangency condition and the requirement (C1). The problem is to analyze
when ψl0 possesses the property (C2). For this purpose one can directly apply
the method used above in the proof of Theorem 2 with some minor changes, for
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 373
V.O. Gorkavyy and O.M. Nevmerzhitska
instance, the standard trigonometric functions have to be replaced by their hyper-
bolic counterparts. It turns out that the key difference between the spherical and
hyperbolic cases is that in Sn×R1 (as well as in Sn) there is an exceptional class
of intrinsically flat ruled surfaces different from vertical cylinders (see items 1 of
Props. 2–3), whereas Hn × R1 and Hn do not admit the ruled surfaces of this
kind. So the following statements hold.
Theorem 3. Let F 2 be a ruled surface in Hn ×R1. The map ψl0 : F 2 → F 2
does not represent pseudospherical congruences for any l0.
Corollary. Let F 2 be a ruled surface in Hn. The map ψl0 : F 2 → F 2 does
not represent pseudospherical congruences for any l0.
7. Conclusion
Thus, if we consider ruled surfaces in Rn, Sn, Hn, Sn × R1, Hn × R1, then
pseudospherical congruences may be represented only by the intrinsically flat
surfaces with nonzero constant extrinsic curvature in Sn × R1 and Sn. This
class of pseudospherical congruences is rather exceptional. So, it would be very
interesting to construct a more general theory of pseudospherical congruences in
Sn ×R1 and Hn ×R1 dealing with nonruled surfaces.
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