Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$
We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by $K$ the nonzero constant Gauss – Kronecker curvature of hypersurfaces, we obtain some characterizations of the...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507833625214976 |
|---|---|
| author | Shu, Shichang Zhu, Tianmin Шу, Шичанґ Чжу, Тяньмін |
| author_facet | Shu, Shichang Zhu, Tianmin Шу, Шичанґ Чжу, Тяньмін |
| author_sort | Shu, Shichang |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:32:19Z |
| description | We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by $K$ the nonzero constant Gauss – Kronecker
curvature of hypersurfaces, we obtain some characterizations of the Riemannian products $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$
$a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$
or
$S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$
$a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$.
|
| first_indexed | 2026-03-24T02:15:36Z |
| format | Article |
| fulltext |
UDC 514
Shichang Shu (School Math. and Inform. Sci., Xianyang Normal Univ., China),
Tianmin Zhu (School Math. and Inform. Sci., Weinan Normal Univ., China)
HYPERSURFACES WITH NONZERO CONSTANT
GAUSS – KRONECKER CURVATURE IN \bfitM \bfitn +\bfone (\pm \bfone ) *
ГIПЕРПОВЕРХНI З НЕНУЛЬОВОЮ СТАЛОЮ КРИВИЗНОЮ
ГАУССА – КРОНЕКЕРА В \bfitM \bfitn +\bfone (\pm \bfone )
We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature
and two distinct principal curvatures one of which is simple. Denoting by K the nonzero constant Gauss – Kronecker
curvature of hypersurfaces, we obtain some characterizations of the Riemannian products Sn - 1(a) \times S1(
\surd
1 - a2),
a2 = 1/(1 +K
2
n - 2 ) or Sn - 1(a)\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1).
Вивчаються гiперповерхнi в одиничнiй сферi та в гiперболiчному просторi з ненульовою сталою кривизною Га-
усса – Кронекера та двома рiзними головними кривизнами, одна з яких є простою. Якщо K — нульова стала
кривизна Гаусса – Кронекера гiперповерхонь, то деякi характеристики рiманових добуткiв можна отримати у вигля-
дi Sn - 1(a)\times S1(
\surd
1 - a2), a2 = 1/(1 +K
2
n - 2 ) або Sn - 1(a)\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1).
1. Introduction. Let Mn be an n-dimensional immersed hypersurface in a real space form Mn+1(c),
c = \pm 1. If c = 1 or c = - 1, we call Mn+1(c) a unit sphere or a hyperbolic space. We notice
that there are many important rigidity results for hypersurfaces with constant mean curvature and two
distinct principal curvatures, see [1, 9], or with constant scalar curvature and two distinct principal
curvatures, see [5, 6]. Since the Gauss – Kronecker curvature of Mn is also an important rigidity
invariant under the isometric immersion, it is natural for us to ask such a question: if the nonzero
Gauss – Kronecker curvature is constant, can we obtain any rigidity results?
In this note we try to study hypersurfaces in Mn+1(c) (c = \pm 1) with nonzero constant Gauss –
Kronecker curvature and two distinct principal curvatures one of which is simple. We introduce the
well-known standard models of complete hypersurfaces with constant Gauss – Kronecker curvature
in Mn+1(c), c = \pm 1.
When c = 1, we consider the standard immersions Sn - k(
\surd
1 - a2) \lhook \rightarrow Rn - k+1 and Sk(a) \lhook \rightarrow
\lhook \rightarrow Rk+1, where 0 < a < 1, 1 \leq k \leq n - 1, and take the Riemannian product immersion
Sk(a)\times Sn - k(
\surd
1 - a2) \lhook \rightarrow Sn+1(c) \subset Rn+2, then it has two distinct constant principal curvatures
\lambda 1 = . . . = \lambda k =
\surd
1 - a2
a
, \lambda k+1 = . . . = \lambda n = - a\surd
1 - a2
,
respectively. We easily see that the Riemannian product Sk(a)\times Sn - k(
\surd
1 - a2) has constant Gauss –
Kronecker curvature K =
\Biggl( \surd
1 - a2
a
\Biggr) k \biggl(
- a\surd
1 - a2
\biggr) n - k
. The square of the norm of the second
fundamental form and the mean curvature of Sn - 1(a)\times S1(
\surd
1 - a2), where a2 = 1/(1 +K
2
n - 2 ),
are
* Project supported by NSF of Shaanxi Province (№ SJ08A31) and NSF of Shaanxi Educational Department
(№ 11JK0479).
c\bigcirc SHICHANG SHU, TIANMIN ZHU, 2016
1540 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1541
| A| 2 = (n - 1)K
1
n - 2 +K
- 2
n - 2 ,
H =
1
n
\{ (n - 1)| K|
1
n - 2 - | K| -
1
n - 2 \} .
When c = - 1, we consider the standard immersions Hn - k( -
\surd
1 + a2) \lhook \rightarrow Rn - k+1
1 and
Sk(a) \lhook \rightarrow Rk+1, where a > 0, 1 \leq k \leq n - 1, and take the Riemannian product immersion
Sk(a) \times Hn - k( -
\surd
1 + a2) \lhook \rightarrow Hn+1(c) \subset Rn+2
1 , then it has two distinct constant principal curva-
tures
\lambda 1 = . . . = \lambda k =
\surd
1 + a2
a
, \lambda k+1 = . . . = \lambda n =
a\surd
1 + a2
,
respectively. We easily see that the Riemannian product Sk(a) \times Hn - k( -
\surd
1 + a2) has con-
stant Gauss – Kronecker curvature K =
\Biggl( \surd
1 + a2
a
\Biggr) k \biggl(
a\surd
1 + a2
\biggr) n - k
. The square of the norm
of the second fundamental form and the mean curvature of Sn - 1(a) \times H1( -
\surd
1 + a2), where
a2 = 1/(K
2
n - 2 - 1), K > 1, and S1(a) \times Hn - 1( -
\surd
1 + a2), where a2 = K
2
n - 2 /(1 - K
2
n - 2 ),
K < 1, are
| A| 2 = (n - 1)K
2
n - 2 +K
- 2
n - 2 ,
H =
1
n
\{ (n - 1)K
1
n - 2 +K
- 1
n - 2 \} .
We obtain some characterizations of Sn - 1(a)\times S1(
\surd
1 - a2), a2 = 1/(1 +K
2
n - 2 ) and Sn - 1(a)\times
\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1):
Theorem 1.1. Let Mn be an n-dimensional with n \geq 3 complete smooth connected and oriented
hypersurface in Mn+1(c) (c = \pm 1) with nonzero constant Gauss – Kronecker curvature K and two
distinct principal curvatures one of which is simple.
(1) When c = 1, K < 0, if
| A| 2 \leq (n - 1)K
2
n - 2 +K
- 2
n - 2 ,
or
| A| 2 \geq (n - 1)K
2
n - 2 +K
- 2
n - 2 ,
then Mn is isometric to the Riemannian product Sn - 1(a)\times S1(
\surd
1 - a2), a2 = 1/(1 +K
2
n - 2 ).
(2) When c = - 1, K > 1, if
| A| 2 \leq (n - 1)K
2
n - 2 +K
- 2
n - 2 ,
or
| A| 2 \geq (n - 1)K
2
n - 2 +K
- 2
n - 2 ,
then Mn is isometric to the Riemannian product Sn - 1(a)\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1542 SHICHANG SHU, TIANMIN ZHU
Theorem 1.2. Let Mn be an n-dimensional with n \geq 3 complete smooth connected and oriented
hypersurface in Mn+1(c) (c = \pm 1) with nonzero constant Gauss – Kronecker curvature K and two
distinct principal curvatures one of which is simple.
(1) When c = 1, K < 0, if
H \leq 1
n
\{ (n - 1)| K|
1
n - 2 - c| K| -
1
n - 2 \} ,
or
H \geq 1
n
\{ (n - 1)| K|
1
n - 2 - c| K| -
1
n - 2 \} ,
then Mn is isometric to the Riemannian product Sn - 1(a)\times S1(
\surd
1 - a2), a2 = 1/(1 +K
2
n - 2 ).
(2) When c = - 1, K > 1, if
H \leq 1
n
\{ (n - 1)| K|
1
n - 2 - c| K| -
1
n - 2 \} ,
or
H \geq 1
n
\{ (n - 1)| K|
1
n - 2 - c| K| -
1
n - 2 \} ,
then Mn is isometric to the Riemannian product Sn - 1(a)\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1).
2. Preliminaries. Let Mn+1(c) be an (n+1)-dimensional connected Riemannian manifold with
constant sectional curvature c (c = \pm 1). Let Mn be an n-dimensional complete smooth connected
and oriented hypersurface in Mn+1(c). We choose a local orthonormal frame e1, . . . , en+1 in
Mn+1(c) such that e1, . . . , en are tangent to Mn . Let \omega 1, . . . , \omega n+1 be the dual coframe. We use
the following convention on the range of indices:
1 \leq A,B,C, . . . \leq n+ 1, 1 \leq i, j, k, . . . \leq n.
The structure equations of Mn+1(c) are given by
d\omega A =
\sum
B
\omega AB \wedge \omega B, \omega AB + \omega BA = 0,
d\omega AB =
\sum
C
\omega AC \wedge \omega CB +\Omega AB, \Omega AB = - 1
2
\sum
C,D
KABCD\omega C \wedge \omega D,
KABCD = c(\delta AC\delta BD - \delta AD\delta BC),
where \Omega AB and KABCD denote the curvature form and the components of the curvature tensor of
Mn+1(c), respectively.
Restricting to Mn,
\omega n+1 = 0, (2.1)
\omega n+1i =
\sum
j
hij\omega j , hij = hji, (2.2)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1543
where hij denotes the components of the second fundamental form of Mn . The structure equations
of Mn are
d\omega i =
\sum
j
\omega ij \wedge \omega j , \omega ij + \omega ji = 0,
d\omega ij =
\sum
k
\omega ik \wedge \omega kj +\Omega ij , \Omega ij = - 1
2
\sum
k,l
Rijkl\omega k \wedge \omega l, (2.3)
Rijkl = c(\delta ik\delta jl - \delta il\delta jk) + (hikhjl - hilhjk), (2.4)
where \Omega ij and Rijkl denote the curvature form and the components of the curvature tensor of Mn,
respectively. From (2.4), we have
n(n - 1)(r - c) = n2H2 - | A| 2,
where n(n - 1)r = R is the scalar curvature, H is the mean curvature and | A| 2 is the squared norm
of the second fundamental form of Mn .
The function K = \mathrm{d}\mathrm{e}\mathrm{t}(hij) is called the Gauss – Kronecker curvature of Mn . We choose
e1, . . . , en such that hij = \lambda i\delta ij , then we see that K = \mathrm{d}\mathrm{e}\mathrm{t} (hij) = \lambda 1\lambda 2 . . . \lambda n . From (2.2) we
obtain
\omega n+1i = \lambda i\omega i, i = 1, 2, . . . , n.
Hence, we get from the structure equations of Mn,
d\omega n+1i = d\lambda i \wedge \omega i + \lambda id\omega i = d\lambda i \wedge \omega i + \lambda i
\sum
j
\omega ij \wedge \omega j . (2.5)
On the other hand, we have on the curvature forms of Mn+1(c),
\Omega n+1i = - 1
2
\sum
C,D
Kn+1iCD\omega C \wedge \omega D =
= - 1
2
\sum
C,D
c(\delta n+1C\delta iD - \delta n+1D\delta iC)\omega C \wedge \omega D = - c\omega n+1 \wedge \omega i = 0.
Therefore, from the structure equations of Mn+1(c), we obtain
d\omega n+1i =
\sum
j
\omega n+1j \wedge \omega ji + \omega n+1n+1 \wedge \omega n+1i +\Omega n+1i =
\sum
j
\lambda j\omega ij \wedge \omega j . (2.6)
From (2.5) and (2.6), we get
d\lambda i \wedge \omega i +
\sum
j
(\lambda i - \lambda j)\omega ij \wedge \omega j = 0. (2.7)
Putting
\psi ij = (\lambda i - \lambda j)\omega ij , (2.8)
we have \psi ij = \psi ji . Hence (2.7) can be written as
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1544 SHICHANG SHU, TIANMIN ZHU\sum
j
(\psi ij + \delta ijd\lambda j) \wedge \omega j = 0.
By E. Cartan’s lemma, we get
\psi ij + \delta ijd\lambda j =
\sum
k
Qijk\omega k, (2.9)
where Qijk are uniquely determined functions such that
Qijk = Qikj .
3. Proofs of theorems. The following Proposition 3.1 original due to Otsuki [7] is useful.
Proposition 3.1. Let Mn be a hypersurface in a real space form Mn+1(c) (c = \pm 1) such that
the multiplicities of the principal curvatures are constant. Then the distribution of the space of the
principal vectors corresponding to each principal curvature is completely integrable. In particular,
if the multiplicity of a principal curvature is greater than 1, then this principal curvature is constant
on each integral submanifold of the corresponding distribution of the space of the principal vectors.
Let Mn be an n-dimensional complete smooth connected and oriented hypersurface with two
distinct principal curvatures one of which is simple and n \geq 3, that is, without loss of generality, we
may assume
\lambda 1 = \lambda 2 = . . . = \lambda n - 1 = \lambda , \lambda n = \mu ,
where \lambda i for i = 1, 2, . . . , n are the principal curvatures of Mn . Thus, we have
K = \lambda n - 1\mu .
From K \not = 0, we conclude that \lambda \not = 0. By changing the orientation for Mn and renumbering
e1, . . . , en if necessary, we may assume that \lambda > 0. Thus
\mu =
K
\lambda n - 1
, (3.1)
0 \not = \lambda - \mu =
\lambda n - K
\lambda n - 1
. (3.2)
We denote the integral submanifold through x \in Mn corresponding to \lambda by Mn - 1
1 (x). Putting
d\lambda =
n\sum
k=1
\lambda ,k \omega k, d\mu =
n\sum
k=1
\mu ,k \omega k,
from Proposition 3.1, we get
\lambda ,1= \lambda ,2= . . . = \lambda ,n - 1= 0 on Mn - 1
1 (x). (3.3)
From (3.1), we obtain
d\mu = - (n - 1)K
\lambda n
d\lambda . (3.4)
Thus, we also have
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1545
\mu ,1= \mu ,2= . . . = \mu ,n - 1= 0 on Mn - 1
1 (x). (3.5)
In this case, we may consider locally \lambda as a function of the arc length s of the integral curve of the
principal vector field en corresponding to the principal curvature \mu . From (2.9) and (3.3), we get,
for 1 \leq j \leq n - 1,
\lambda ,n \omega n =
n\sum
i=1
\lambda ,i \omega i = d\lambda = d\lambda j =
n\sum
k=1
Qjjk\omega k =
n - 1\sum
k=1
Qjjk\omega k +Qjjn\omega n.
Therefore, we obtain
Qjjk = 0, 1 \leq k \leq n - 1, and Qjjn = \lambda ,n . (3.6)
By (2.9) and (3.5), we have
\mu ,n \omega n =
n\sum
i=1
\mu ,i \omega i = d\mu = d\lambda n =
n\sum
k=1
Qnnk\omega k =
n - 1\sum
k=1
Qnnk\omega k +Qnnn\omega n.
Hence, we obtain
Qnnk = 0, 1 \leq k \leq n - 1, and Qnnn = \mu ,n . (3.7)
From (3.4), we get
Qnnn = \mu ,n= - (n - 1)K
\lambda n
\lambda ,n .
From the definition of \psi ij , if i \not = j, we have \psi ij = 0 for 1 \leq i \leq n - 1 and 1 \leq j \leq n - 1.
Therefore, from (2.9), if i \not = j and 1 \leq i \leq n - 1 and 1 \leq j \leq n - 1 we obtain
Qijk = 0 for any k. (3.8)
By (2.9), (3.6), (3.7) and (3.8), for j < n, we get
\psi jn =
n\sum
k=1
Qjnk\omega k = Qjjn\omega j +Qjnn\omega n = \lambda ,n \omega j . (3.9)
From (2.8), (3.2) and (3.9), for j < n, we obtain
\omega jn =
\psi jn
\lambda - \mu
=
\lambda ,n
\lambda - \mu
\omega j =
\lambda n - 1\lambda ,n
\lambda n - K
\omega j .
Thus, from the structure equations of Mn we have
d\omega n =
n - 1\sum
k=1
\omega k \wedge \omega kn + \omega nn \wedge \omega n = 0.
Therefore, we may put \omega n = ds. By (3.3), we get
d\lambda = \lambda ,n ds, \lambda ,n=
d\lambda
ds
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1546 SHICHANG SHU, TIANMIN ZHU
Thus, we obtain
\omega jn =
\lambda n - 1 d\lambda
ds
\lambda n - K
\omega j =
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K|
1
n )
ds
\omega j . (3.10)
From (3.10) and the structure equations of Mn+1(c), for j < n, we have
d\omega jn =
n - 1\sum
k=1
\omega jk \wedge \omega kn + \omega jn \wedge \omega nn + \omega jn+1 \wedge \omega n+1n +\Omega jn =
=
n - 1\sum
k=1
\omega jk \wedge \omega kn + \omega jn+1 \wedge \omega n+1n - c\omega j \wedge \omega n =
=
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
n - 1\sum
k=1
\omega jk \wedge \omega k - (\lambda \mu + c)\omega j \wedge ds.
Differentiating (3.10), we obtain
d\omega jn =
d2(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds2
ds \wedge \omega j +
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
d\omega j =
=
d2(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds2
ds \wedge \omega j +
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
n\sum
k=1
\omega jk \wedge \omega k =
=
\left\{ - d
2(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds2
+
\Biggl[
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
\Biggr] 2\right\} \omega j \wedge +ds
+
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
n - 1\sum
k=1
\omega jk \wedge \omega k.
From the previous two equalities, we get
d2(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds2
-
\Biggl\{
d(\mathrm{l}\mathrm{o}\mathrm{g} | \lambda n - K| 1/n)
ds
\Biggr\} 2
- (\lambda \mu + c) = 0. (3.11)
If we define \varpi = | \lambda n - K| - 1/n, from (3.11) we have
d2\varpi
ds2
+\varpi (\lambda \mu + c) = 0. (3.12)
On the other hand, from (3.10), we have \nabla enen =
\sum n
i=1
\omega ni(en)ei = 0. By the definition of
geodesic, we know that any integral curve of the principal vector field corresponding to the principal
curvature \mu is a geodesic. Thus, we see that \varpi (s) is a function defined in ( - \infty ,+\infty ) since Mn is
complete and any integral curve of the principal vector field corresponding to \mu is a geodesic.
We can prove the following lemma.
Lemma 3.1. If c = 1 or c = - 1 and K > 1, then the positive function \varpi is bounded from
above.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1547
Proof. From (3.2), we know that \lambda n - K \not = 0. Thus (3.1) and (3.12) imply that
d2\varpi
ds2
+\varpi
c\lambda n - 2 +K
\lambda n - 2
= 0, (3.13)
that is
d2\varpi
ds2
+\varpi
\Bigl[
c+K(K \pm \varpi - n)2/n - 1
\Bigr]
= 0. (3.14)
Multiplying (3.14) by 2
d\varpi
ds
and integrating, we get
\biggl(
d\varpi
ds
\biggr) 2
+ c\varpi 2 +\varpi 2(K \pm \varpi - n)2/n = C,
where C is a constant. Thus, we obtain
c+ (K \pm \varpi - n)2/n \leq C
\varpi 2
. (3.15)
If the positive function \varpi is not bounded from above, that is, \varpi \rightarrow +\infty . From (3.15), we have that
c+K2/n \leq 0, a contradiction with the assumption. Thus we conclude.
We can also prove the following lemma.
Lemma 3.2. (1) Let
PK(t) = ct
n - 2
n +K, t > 0.
(i) For c = 1, if K < 0, then PK(t) is a strictly monotone increasing function of t and has a
positive real root t0 = ( - K)
n
n - 2 .
(ii) For c = - 1, if K > 0, then PK(t) is a strictly monotone decreasing function of t and has
a positive real root t0 = K
n
n - 2 .
(2) Let
| A| 2(t) = 1
t2(n - 1)/n
\{ (n - 1)t2 +K2\} , t > 0,
t0 = ( - K)
n
n - 2 for c = 1, K < 0 and t0 = K
n
n - 2 for c = - 1, K > 0. Then
(i) If t \geq | K| and t0 \geq | K| , then t \leq t0 holds if and only if | A| 2(t) \leq (n - 1)t
2/n
0 + t
- 2/n
0 and
t \geq t0 holds if and only if | A| 2(t) \geq (n - 1)t
2/n
0 + t
- 2/n
0 .
(ii) If t \leq | K| and t0 \leq | K| , then t \leq t0 holds if and only if | A| 2(t) \geq (n - 1)t
2/n
0 + t
- 2/n
0 and
t \geq t0 holds if and only if | A| 2(t) \leq (n - 1)t
2/n
0 + t
- 2/n
0 .
(3) Let
H(t) =
1
nt(n - 1)/n
\{ (n - 1)t+K\} , t > 0.
Then
(i) For c = 1, K < 0 and t0 = ( - K)
n
n - 2 , if t \geq K, then t \leq t0 holds if and only if
H(t) \leq 1
n
\{ (n - 1)t
1/n
0 - t
- 1/n
0 \} and t \geq t0 holds if and only if H(t) \geq 1
n
\{ (n - 1)t
1/n
0 - t
- 1/n
0 \} .
(ii) For c = - 1, K > 0 and t0 = K
n
n - 2 ,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1548 SHICHANG SHU, TIANMIN ZHU
(a) if t \geq K and t0 \geq K, then t \leq t0 holds if and only if H(t) \leq 1
n
\{ (n - 1)t
1/n
0 + t
- 1/n
0 \} and
t \geq t0 holds if and only if H(t) \geq 1
n
\{ (n - 1)t
1/n
0 + t
- 1/n
0 \} ;
(b) if t \leq K and t0 \leq K, then t \leq t0 holds if and only if H(t) \geq 1
n
\{ (n - 1)t
1/n
0 + t
- 1/n
0 \} and
t \geq t0 holds if and only if H(t) \leq 1
n
\{ (n - 1)t
1/n
0 + t
- 1/n
0 \} .
Proof. (1) Obvious fact.
(2) We have
d| A| 2(t)
dt
=
2(n - 1)t(2 - 3n)/n
n
(t2 - K2),
it follows that the solution of
d| A| 2(t)
dt
= 0 is t = | K| . Therefore, we know that t \leq | K| if and
only if | A| 2(t) is a decreasing function, t \geq | K| if and only if | A| 2(t) is an increasing function and
| A| 2(t) obtain its minimum at t = | K| .
If t0 \geq | K| , since t \geq | K| if and only if | A| 2(t) is an increasing function, we infer that if
t \geq | K| , then t \leq t0 holds if and only if
| A| 2(t) \leq | A| 2(t0) =
1
t
2(n - 1)/n
0
\{ (n - 1)t20 +K2\} =
=
1
t
2(n - 1)/n
0
\Biggl\{
(n - 1)t20 +
\biggl[ \biggl(
ct
n - 2
n
0 +K
\biggr)
- ct
n - 2
n
0
\biggr] 2\Biggr\}
=
=
1
t
2(n - 1)/n
0
\Biggl\{
(n - 1)t20 +
\biggl[
- ct
n - 2
n
0
\biggr] 2\Biggr\}
= (n - 1)t
2/n
0 + t
- 2/n
0 ,
where c = 1, t0 = ( - K)
n
n - 2 and K < 0 or c = - 1, t0 = K
n
n - 2 and K > 0. By the same reason,
the rest of case (2) follows.
(3) Since
dH(t)
dt
=
n - 1
n2t(2n - 1)/n
(t - K),
we see that H(t) is an increasing function if t \geq K and H(t) is a decreasing function if t \leq K,
then it follows the result of (3).
Proof of Theorem 1.1. Putting t = \lambda n(> 0), from (3.1), we see that the square of the norm of
second fundamental form | A| 2 = (n - 1)\lambda 2 +
K2
\lambda 2(n - 1)
=
1
t2(n - 1)/n
\{ (n - 1)t2 + K2\} = | A| 2(t).
From (3.13), we have
d2\varpi
ds2
+\varpi
PK(t)
t
n - 2
n
= 0. (3.16)
(1) When c = 1, we consider two cases t \geq | K| and t \leq | K| .
Case (i). If t \geq | K| , we also consider two subcases | K| > t0 and | K| \leq t0, where t0 =
= ( - K)
n
n - 2 .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1549
If | K| > t0, since t \geq | K| , we get t > t0 . Since we assume that K < 0, by Lemma 3.2, we infer
that PK(t) > PK(t0) = 0. From (3.16), we have
d2\varpi
ds2
< 0, this implies that
d\varpi (s)
ds
is a strictly
monotone decreasing function of s and thus it has at most one zero point for s \in ( - \infty ,+\infty ). If
d\varpi (s)
ds
has no zero point in ( - \infty ,+\infty ), then \varpi (s) is a monotone function of s in ( - \infty ,+\infty ). If
d\varpi (s)
ds
has exactly one zero point s0 in ( - \infty ,+\infty ), then \varpi (s) is a monotone function of s in both
( - \infty , s0] and [s0,+\infty ).
On the other hand, from Lemma 3.1, we know that \varpi (s) is bounded. Since \varpi (s) is bounded
and monotonic when s tends to infinity, we know that both \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow - \infty \varpi (s) and \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow +\infty \varpi (s) exist
and then we get
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow - \infty
d\varpi (s)
ds
= \mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow +\infty
d\varpi (s)
ds
= 0. (3.17)
This is impossible because
d\varpi (s)
ds
is a strictly monotone decreasing function of s. Therefore, we
know that the case | K| > t0 does not occur. It follows that | K| \leq t0 .
If | K| \leq t0, since t \geq | K| , from Lemma 3.2 and (3.16), we have
| A| 2 \leq (n - 1)K
2
n - 2 +K
- 2
n - 2 = (n - 1)t
2/n
0 + t
- 2/n
0 ,
holds if and only if t \leq t0 if and only if PK(t) \leq 0 and if and only if
d2\varpi
ds2
\geq 0. Also
| A| 2 \geq (n - 1)K
2
n - 2 +K
- 2
n - 2 = (n - 1)t
2/n
0 + t
- 2/n
0 ,
holds if and only if t \geq t0 if and only if PK(t) \geq 0 and if and only if
d2\varpi
ds2
\leq 0. Thus
d\varpi
ds
is
a monotonic function of s \in ( - \infty ,+\infty ), this implies that
d\varpi (s)
ds
has at most one zero point for
s \in ( - \infty ,+\infty ). If
d\varpi (s)
ds
has no zero point in ( - \infty ,+\infty ), then \varpi (s) is a monotone function of
s in ( - \infty ,+\infty ). If
d\varpi (s)
ds
has exactly one zero point s0 in ( - \infty ,+\infty ), then \varpi (s) is a monotone
function of s in both ( - \infty , s0] and [s0,+\infty ). Therefore, we see that \varpi (s) is monotonic when s
tends to infinity. Since \varpi (s) is bounded and monotonic when s tends to infinity, we know that both
\mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow - \infty \varpi (s) and \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow +\infty \varpi (s) exist and (3.17) holds. From the monotonicity of
d\varpi (s)
ds
, we
have
d\varpi (s)
ds
\equiv 0 and \varpi (s) = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}. Combining \varpi = | \lambda n - K| - 1/n and (3.1), we conclude that \lambda
and \mu are constant, that is, Mn is isoparametric. From the classical result of Cartan [3] (see also [4,
p. 238]), we know that Mn is isometric to the Riemannian product Sn - 1(a)\times S1(
\surd
1 - a2), where
a2 = 1/(1 +K
2
n - 2 ).
Case (ii). If t \leq | K| , we consider two subcases | K| < t0 and | K| \geq t0, where t0 = ( - K)
n
n - 2 .
If | K| < t0, since t \leq | K| , we have t < t0 . From K < 0 and Lemma 3.2, it follows that
PK(t) < PK(t0) = 0. From (3.16), we have
d2\varpi
ds2
> 0. Thus
d\varpi (s)
ds
is a strictly monotone
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1550 SHICHANG SHU, TIANMIN ZHU
increasing function of s. By the same arguments as in case (i), we conclude that | K| < t0 does not
occur, then | K| \geq t0 .
If | K| \geq t0, since t \leq | K| , from Lemma 3.2 and (3.16), we have that | A| 2 \leq (n - 1)t
2/n
0 + t
- 2/n
0
holds if and only if t \geq t0 if and only if PK(t) \geq 0 and if and only if
d2\varpi
ds2
\leq 0. Also | A| 2 \geq
\geq (n - 1)t
2/n
0 + t
- 2/n
0 holds if and only if t \leq t0 if and only if PK(t) \leq 0 and if and only if
d2\varpi
ds2
\geq 0. By the same arguments as in the proof of case (i), we conclude.
(2) When c = - 1, we also consider two cases t \geq | K| and t \leq | K| .
Case (i). If t \geq | K| , we consider two subcases | K| > t0 and | K| \leq t0, where t0 = K
n
n - 2 .
If | K| > t0, since t \geq | K| , we have t > t0 . Since we assume that K > 1, by Lemma 3.2,
we infer that PK(t) < PK(t0) = 0. From (3.16), we have
d2\varpi
ds2
> 0, this implies that
d\varpi (s)
ds
is a
strictly monotone increasing function of s and thus it has at most one zero point for s \in ( - \infty ,+\infty ).
By use of the same method as in the proof of case (i) in (1), we know that the case | K| > t0 does
not occur. It follows that | K| \leq t0 .
If | K| \leq t0, since t \geq | K| , from Lemma 3.2 and (3.16), we see that
| A| 2 \leq (n - 1)K
2
n - 2 +K
- 2
n - 2 = (n - 1)t
2/n
0 + t
- 2/n
0 ,
holds if and only if t \leq t0 if and only if PK(t) \geq 0 and if and only if
d2\varpi
ds2
\leq 0. Also
| A| 2 \geq (n - 1)K
2
n - 2 +K
- 2
n - 2 = (n - 1)t
2/n
0 + t
- 2/n
0 ,
holds if and only if t \geq t0 if and only if PK(t) \leq 0 and if and only if
d2\varpi
ds2
\geq 0. Thus
d\varpi
ds
is a
monotonic function of s \in ( - \infty ,+\infty ). By use of the same method as in the proof of case (i) in
(1), we conclude that \lambda and \mu are constant, that is, Mn is isoparametric. From the classical result
of Cartan [2] (see also [8] or [4, p. 238]), we know that Mn is isometric to the Riemannian product
Sn - 1(a)\times H1( -
\surd
1 + a2), a2 = 1/(K
2
n - 2 - 1).
Case (ii). If t \leq | K| , we consider two subcases | K| < t0 and | K| \geq t0, where t0 = K
n
n - 2 .
If | K| < t0, since t \leq | K| , we have t < t0 . From K > 1 and Lemma 3.2, it follows that
PK(t) > PK(t0) = 0. From (3.16), we have
d2\varpi
ds2
< 0. Thus
d\varpi (s)
ds
is a strictly monotone
decreasing function of s. By the same arguments as in case (i) of (1), we conclude that | K| < t0
does not occur, then | K| \geq t0 .
If | K| \geq t0, since t \leq | K| , from Lemma 3.2 and (3.16), we have that | A| 2 \leq (n - 1)t
2/n
0 + t
- 2/n
0
holds if and only if t \geq t0 if and only if PK(t) \leq 0 and if and only if
d2\varpi
ds2
\geq 0. Also | A| 2 \geq
\geq (n - 1)t
2/n
0 + t
- 2/n
0 holds if and only if t \leq t0 if and only if PK(t) \geq 0 and if and only if
d2\varpi
ds2
\leq 0. By the same arguments as in the proof of case (i) of (2), we conclude.
Theorem 1.1 is proved.
Proof of Theorem 1.2. (1) When c = 1, since we assume that K < 0 and t = \lambda n(> 0), we see
that t > K . From t0 > K, where t0 = ( - K)
n
n - 2 , Lemma 3.2 and (3.16), we see that
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
HYPERSURFACES WITH NONZERO CONSTANT GAUSS – KRONECKER CURVATURE IN Mn+1(\pm 1) 1551
H \leq 1
n
\{ (n - 1)| K|
1
n - 2 - | K| -
1
n - 2 \} =
1
n
\{ (n - 1)t
1/n
0 - t
- 1/n
0 \} ,
holds if and only if t \leq t0 if and only if PK(t) \leq 0 and if and only if
d2\varpi
ds2
\geq 0. Also
H \geq 1
n
\{ (n - 1)| K|
1
n - 2 - | K| -
1
n - 2 \} =
1
n
\{ (n - 1)t
1/n
0 - t
- 1/n
0 \} ,
holds if and only if t \geq t0 if and only if PK(t) \geq 0 and if and only if
d2\varpi
ds2
\leq 0. Thus
d\varpi
ds
is
a monotonic function of s \in ( - \infty ,+\infty ). By use of the same method as in the proof of (1) in
Theorem 1.1, we know that Mn is isometric to the Riemannian product Sn - 1(a) \times S1(
\surd
1 - a2),
where a2 = 1/(1 +K
2
n - 2 ).
(2) When c = - 1, since we assume that K > 1, we consider two cases t \geq K and t \leq K . By
Lemma 3.2, we see that the function \varpi is bounded. It suffices to use the same method as in the proof
of (2) in Theorem 1.1.
Theorem 1.2 is proved.
References
1. Alias L. J., de Almeida S.C., Brasil A. (Jr.) Hypersurfaces with constant mean curvature and two principal curvatures
in Sn+1 // An. Acad. Brasil. ciênc. – 2004. – 76. – P. 489 – 497.
2. Cartan E. Familles de surfaces isoparamétriques dans les espaces \'a courbure constante // Ann. Mat. – 1938. – 17. –
P. 177 – 191.
3. Cartan E. Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques // Math. Z. –
1939. – 45. – P. 335 – 367.
4. Cecil T., Ryan P. Tight and taut immersions of manifolds // Res. Notes Math. – 1985. – 107.
5. Cheng Q.-M., Shu S. C., Suh Y. J. Compact hypersurfaces in a unit sphere // Proc. Roy. Soc. Edinburgh A. – 2005. –
135. – P. 1129 – 1137.
6. Hu Z., Zhai S. Hypersurfaces of the hyperbolic space with constant scalar curvature // Results Math. – 2005. – 48. –
P. 65 – 88.
7. Otsuki T. Minimal hypersurfaces in a Riemannian manifold of constant curvature // Amer. J. Math. – 1970. – 92. –
P. 145 – 173.
8. Ryan P. J. Hypersurfaces with parallel Ricci tensor // Osaka J. Math. – 1971. – 8. – P. 251 – 259.
9. Wei G. Complete hypersurfaces with constant mean curvature in a unit sphere // Monatsh. Math. – 2006. – 149. –
S. 251 – 258.
Received 13.01.13,
after revision — 30.08.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
|
| id | umjimathkievua-article-1940 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:36Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a2/45c7c964c8baa07450e5f8e047daa4a2.pdf |
| spelling | umjimathkievua-article-19402019-12-05T09:32:19Z Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ Гiперповерхнi з ненульовою сталою кривизною Гаусса–Кронекера в $M^{n+1}(±1)$ Shu, Shichang Zhu, Tianmin Шу, Шичанґ Чжу, Тяньмін We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by $K$ the nonzero constant Gauss – Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$ $a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$ or $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$. Вивчаються гiперповерхнi в одиничнiй сферi та в гiперболiчному просторi з ненульовою сталою кривизною Гаусса – Кронекера та двома рiзними головними кривизнами, одна з яких є простою. Якщо $K$ — нульова стала кривизна Гаусса – Кронекера гiперповерхонь, то деякi характеристики рiманових добуткiв можна отримати у виглядi $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$ $a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$ або $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$. Institute of Mathematics, NAS of Ukraine 2016-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1940 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 11 (2016); 1540-1551 Український математичний журнал; Том 68 № 11 (2016); 1540-1551 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1940/922 Copyright (c) 2016 Shu Shichang; Zhu Tianmin |
| spellingShingle | Shu, Shichang Zhu, Tianmin Шу, Шичанґ Чжу, Тяньмін Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title | Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title_alt | Гiперповерхнi з ненульовою сталою кривизною
Гаусса–Кронекера в $M^{n+1}(±1)$ |
| title_full | Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title_fullStr | Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title_full_unstemmed | Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title_short | Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ |
| title_sort | hypersurfaces with nonzero constant gauss – kronecker curvature in $m^{n+1}(±1)$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1940 |
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