# hyperbolic cylinder

## hyperbolic cylinder

[¦hī·pər¦bäl·ik ′sil·ən·dər]
(mathematics)
A cylinder whose directrix is a hyperbola.
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If inf[([lambda] - [mu]).sup.2] > 0, Cao-Wei [2] proved that M is isometric to hyperbolic cylinder [H.sup.m] (-n/m) x [H.sup.n-m], (-n/n-m), 1 [less than or equal to] m [less than or equal to] n - 1.
Although Wu [13] pointed out that the condition inf [([lambda] - [mu]).sup.2] > 0, either for the case of constant mean curvature or constant scalar curvature, can not be dropped down, by replacing the condition inf[([lambda] - [mu]).sup.2] > 0 with the squared norm of the second fundamental form S [greater than or equal to] (n-i)(2-n-nR)/n-2 or S [less than or equal to] (n-i)(2-n-nR)/n-2 n-2/2-n-nR' Chu-Zhai also proved that M is isometric to hyperbolic cylinder [H.sup.n-1] (nR/n-2) x [H.sup.1] (nR/2-n-nR).
Therefore, the results due to Abe-Koike-Yamaguchi [1] lead to M be an isoparametric hypersurface and isometric to the hyperbolic cylinder [H.sup.n-1] ([c.sub.1]) x [H.sup.1] ([c.sub.2]), where [c.sub.1] < 0, [c.sub.2] < 0, 1/[c.sub.1], 1/[c.sub.2] = 1/c.
However, in the other cases if [phi](t) = [[phi].sub.0] > 0 is a constant and [phi](t) = t, the rotation hypersurface [M.sub.1,T] is the Lorentzian cylinder [S.sup.n-1](0,[[phi].sub.0]) x [L.sup.1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the hyperbolic cylinder [H.sup.n-1] (0,-[[phi].sub.0]) x R, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the pseudo-spherical cylinder [S.sub.1.sup.n-1] (0,[[phi].sub.o])x R.
(2) A rational rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] parametrized by (2.3) has pointwise 1-type Gauss map of the first kind if and only if it is an open portion of a time-like hyperplane or a hyperbolic cylinder [H.sup.n-1] x R of [L.sup.n+1].
For example, space-like hyperplanes, Lorentzian hyperplanes, hyperbolic spaces [H.sup.n](0,-c), de Sitter spaces [S.sub.1.sup.n](0,c), Lorentzian cylinders [S.sup.n-1](0,c) x [L.sup.1], hyperbolic cylinders [H.sup.n-1](0,-c) x R, and the pseudo-spherical cylinders [S.sub.1.sup.n-1] (0,c) x R of [L.sup.n+1] have 1-type Gauss map.
He shows that [M.sup.n] is totally umbilical, or [M.sup.n] is the hyperbolic cylinder in [S.sup.n+p.sub.p](c) or [M.sup.n] has unbounded volume and positive Ricci curvature.
From the equality in lemma 2.1, we know [M.sup.n] isometric to a hyperbolic cylinder [S.sup.n-1](1 - tan[h.sup.2]r) x [H.sup.1](l- [coth.sup.2]r) in [S.sup.n+1.sub.1(1).
In [9], Liu obtained a pinching theorem on space-like hypersurface with constant R, he proved that if n(1 - R) [less than or equal to] sup S [less than or equal to] D(n, R), then either sup S = n(1 - R) and [M.sup.n] is totally umbilical or sup S = D(n, R) and [M.sup.n] is a hyperbolic cylinder, where D(n,R) = n/(n-2)(n-nR-2)[n(n - 1)[(1 - R).sup.2] - 4(n - 1)(1 - R)+n].
If a [not equal to] 0, b < 1 and the sectional curvature of [M.sup.n] is non-negative, then [M.sup.n] is totally umbilical or a hyperbolic cylinder [H.sup.1] (1 - [coth.sup.2]r) x [S.sup.n-1] (1 - [tanh.sup.2]r).
Due to this fact, the hyperbolic punctured disk [D.sup.*] is also called the parabolic cylinder while the hyperbolic annuli A(R) are also called hyperbolic cylinders [8].
Hyperbolic annuli (also known as "hyperbolic cylinders" [8]) have a single modulus and two funnel ends.
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