Tietze-type theorem on 2-dimensional Riemannian manifolds without conjugate points

Abstract

Let A be an open connected subset of a C1 complete simply
connected 2-dimensional Riemannian manifold without conjugate points
W2. The main result of this short article states that: a point x of A has
a local maximal visibility if and only if x is a point of the convex kernel
of A. Thus we obtain a Tietze-type theorem in W2.

Full text

Tietze-type theorem in 2-dimensional Riemannian
manifolds without conjugate points
S. Shenawy
Abstract. Let Abe an open connected subset of a C∞complete simply
connected 2-dimensional Riemannian manifold without conjugate points
W2. The main result of this short article states that: a point xof Ahas
a local maximal visibility if and only if xis a point of the convex kernel
of A. Thus we obtain a Tietze-type theorem in W2.
M.S.C. 2010: 52A10, 52A30, 53B20.
Key words: maximal visibility, starshaped set, kernel, conjugate points, Tietze-type
theorem.
1 Introduction
The first local versus global result involving usual convexity is due to Tietze [10].
Tietze and Nakajima proved that a closed connected locally convex set in Euclidean
space is convex, thus they established a global property from a local one [9, 12, 11, 16,
17]. In [16], the authors obtained similar results in which local convexity was replaced
by weaker conditions called C−convexity and strong local C−convexity. In [12], the
authors proved that under certain conditions a starshaped set is characterized by
the existence of points enjoying a local condition, maximal visibility. Using maximal
visibility, J. Cell presented a similar result for open connected set and for its closure
and he obtained a Tietze-type theorem for partially convex planar set [10]. M. Breen
studied the union of starshaped sets and the union of orthogonally starshaped sets
using the concept of local maximal visibility in the plane [9]. In the present work
we get a Tietze-type theorem for open connected subsets of a C∞complete simply
connected 2-dimensional Riemannian manifold without conjugate points W2.
Now, we introduce some properties of C∞complete simply connected n-dimensional
Riemannian manifold without conjugate points Wn. At first recall that, by the well-
known Hopf-Rinow theorem, if a Riemannian manifold is complete, then it is geodesi-
cally connected. Moreover, any two points pand qcan be joined by a minimal
geodesic. So it is worth pointing out that in order to obtain convexity in Riemannian
manifolds, the assumption of completeness can not be removed [2]. The behavior of
∗
Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 133-137.
c
°Balkan Society of Geometers, Geometry Balkan Press 2011.

134 S. Shenawy
geodesics in manifolds without conjugate or focal points has been discussed by many
geometers as Morse, Hedlund, Green, Eberlein and others [13]. Now let Wnbe a
C∞complete simply connected n-dimensional Riemannian manifold without conju-
gate points. The Euclidean space as well as the Hyperbolic space are examples of
complete Riemannian manifolds without conjugate points. In such Riemannian man-
ifolds Wn, no two geodesics intersect twice due to the absence of conjugate points
and hence for any two different points pand qthere is a unique and hence minimal
closed geodesic segment, denoted by [pq], joining them. This fact implies that the
three types of convexities in complete Riemannian manifolds that were introduced
in [1] are identical in Wnand each of them is equivalent to the classical concept of
convexity in the Euclidean space En. In the following we introduce this classical
concept of convexity in Wn. For more properties of Wnand convex sets in it see
[4, 3, 6, 7, 8, 18, 14, 15].
2 Notations and definitions
Let Abe a subset of Wn. We say that Ais starshaped if there exists a point pin
Asuch that for any point xin Athe closed geodesic segment[px], joining pand x, is
in A. In this case we say that psees xvia A. The set of all such points pis called
the kernel of Aand is denoted by ker A. M. Beltagy proved that kerAis convex for
n= 2[6]. Ais convex if ker A=Ai.e. for each x, y ∈A, the closed geodesic segment
[xy] joining them is contained in Aand hence xsees yvia A. The open and closed
geodesic discs are both convex sets in Wn.
A set Ais said to be locally convex at a point pin Aif there exists a neighborhood
Nof psuch that N∩Ais convex. It is clear that the open set is a locally convex set.
The convex hull of a set Ais the smallest convex set that contains Aand is denoted
by C(A). It is clear that C(A) = Awhen Ais convex. Let Apbe the set of all points
xof Athat psees via A. We say that phas higher visibility via Athan qif Aq⊂Ap.
A point of (local) maximal visibility of Ais a point p∈Asuch that there exists a
neighborhood Nof psatisfying that phas higher visibility than any other point of
N∩A[10].
A geodesic path between two points pand qis the union of nclosed geodesic
segments [xixi+1],0≤i≤n−1 where xi’s are distinct points of Wnwith x0=p
and xn=q. Every geodesic segment [xixi+1 ] is called a side of the geodesic path. A
set Ais called geodesically connected if for each two points xand yin Athere exists
a geodesic path in Ajoining xand y[5]. −→
xy denotes the geodesic ray starting from x
and passing through y, where (xy) denotes the open geodesic segment joining xand
y.
3 Maximal visibility in W2
In this section we present the main theorem of this paper that introduces the as-
sumptions of a subset Aof W2to get a characterization of the kernel of Ausing the
concept of local maximal visibility. We begin with the following two lemmas.
Lemma 3.1. If Ais a nonempty open connected subset of W2, then Ais geodesically
connected.

Tietze-type theorem in W2135
Proof. Let p∈Aand let Apdenote the set of all points in Awhich can be joined
to pby a geodesic path in A. We claim that Apis both open and closed as a subset
of A. To see that Apis open, let q∈Ap. Since Ais locally convex, there exists a
neighborhood Nof qsuch that N∩Ais convex. It follows that each point of N∩A
can be joined to qand hence to pby a geodesic path. Thus N∩A(as an open set
in the relative topology) is a subset of Apand Apis open in A. To see that Apis
closed, let z∈¯
Ap. Since Ais locally convex, there exist a neighborhood Nof zsuch
that N∩Ais convex and is a neighborhood of zin the relative topology on A, and
hence N∩Amust also intersect Apsince z∈¯
Ap. Thus there exists a point win
(N∩A)∩Ap. Since N∩Ais convex, [wz]⊂A. But wcan be joined to pby a
geodesic path in A, hence zcan also. Thus z∈Ap, and Apis closed. Since Apis
closed and open in Aand Ais connected, it follows that Apmust equal to Aand
hence Ais geodesically connected. ¤
Lemma 3.2. Let Abe an open connected subset of W2. If [xy]⊂Aand [yz]⊂A,
then there exists a point qin [xy]with q6=y, such that the convex hull of {q, y , z}is
contained in A.
Proof. Let Kbe the set of all points pin [yz] such that C{q, y, p} ⊂ Afor some q
in [xy] with q6=y. Since y∈K, K is not empty. We claim that Kis both open
and closed in [yz]. Since [yz] is connected, this claim implies that K= [yz] and the
proof is complete. To see that Kis open in [yz], let p∈K. Then there exists a point
q∈[xy) such that C{q, y, p} ⊂ A. Since Ais open and hence locally convex then
there is a neighborhood Nof psuch that N∩Ais convex. Let a∈N∩[pz] and
b∈N∩[pq], then the ray −→
ab meets [yq] at f, and hence C{a, y, f } ⊂ A, since the
convex hull of C{a, b, p}is contained in N∩A⊂Asee Figure 1.
y
q
p
z
x
f
a
b
C{
,y,
}
af
Figure 1: The set Kis open in [yz]
Therefore, a∈Kand consequently pis an interior point in K. To see that K
is closed in [yz], let p∈¯
K. Since Ais locally convex, then there is a neighborhood
Nof psuch that N∩Ais convex. Moreover, Ncontains a point aof Ksuch that

136 S. Shenawy
a∈N∩[yp]. Then there exists a point q∈[xy) such that C{q, y , a} ⊂ A. Choose
b∈N∩[aq], then the ray −→
pb meets [qy) at fsee Figure 2.
y
q
p
z
x
f
ab
C{
p,y,}
f
Figure 2: The set Kis closed in [yz]
Since C{a, q, y} ⊂ Aby definition of Kand C{a, p, b} ⊂ Aby local convexity,
then C{p, f, y} ⊂ C{a, q, y} ∪ C{a, p, b} ⊂ Aand hence p∈K. Then Kis closed
and the proof is complete. ¤
Theorem 3.3. Let Abe an open connected subset of W2. Then the kernel of Ais
the set of all points of maximal visibility.
Proof. Let Vbe the set of all points of maximal visibility in A. We want to prove that
V= ker A. It is clear that ker A⊂V, so we will show that V⊂ker A. Let x /∈ker A
i.e. there is a point yin Asuch that [xy]6⊂ A. By Lemma 3.1, Ais geodesically
connected since Ais an open connected subset of W2. Therefore, there is a geodesic
path with nsides such that x=x0, x1, ..., xn=yand
Un−1
i=0 [xixi+1]⊂A
Choose a geodesic path Pwith minimal nand so Pmust be simple (dose not intersect
itself). Now, the points x0, x1, x2are non-geodesic triple. By Lemma 3.2, there exists
a point tin [x1x2] such that C([x0x1]∪[x1t]) ⊂A. Let Mbe the set of all such points
tin [x1x2]. It is clear that Mis convex, so we get a point zsuch that M= [xz] or
M= [xz). Since Ais open, M= [xz). Now, for any neighborhood Nof x, all points
of N∩[x0x1] see zvia Awhere xdoes not i.e. xis not a point of maximal visibility
in Aand therefore xis not in V. Hence V⊂ker Aand the proof is complete. ¤
Theorem 3.3 is valid in the Euclidean space Enas a manifold without conjugate
points[12]. Also it is valid in the hyperbolic space Hnsince the Beltrami (or central
projection) map defined in [5] takes Hnto Enand preserves geodesics. But the
generalization of Theorem 3.3 to any Wnis more difficult and is left as an open
question.

Tietze-type theorem in W2137
References
[1] S. Alexander, Local and global convexity in complete Riemannian manifolds, Pa-
cific Journal of Mathematics 76(2) (1978), 283–289.
[2] R. Bartolo, A. Germinario, M. Sanchez, Convexity of domains of Riemannian
manifolds, arXiv:math.DG/0004075 v1 (2009).
[3] M. Beltagy, Foot points and convexity in manifolds without focal points, Bull.
Cal. Math. Soc. 82 (1990), 338–348.
[4] M. Beltagy, Immersions into Manifolds without Conjugate Points, PhD thesis,
University of Durham, England, 1982.
[5] M. Beltagy, On starshaped sets, Bull. Malaysian Math. Sci. Soc. 11 (1988), 49–57.
[6] M. Beltagy, Sufficient conditions for convexity in manifolds without focal points,
Comment. Math. Univ. Carolinae 34(1993), 443–449.
[7] M Beltagy, and A. El Araby, Starshaped sets in Riemannian manifolds without
conjugate points, Far East J. Math. Sci. (FJMS) 6 (2002), 187–169.
[8] M. Beltagy, and S. Shenawy, Sets with zero-dimensional kernels, Int. J. of Mod.
Math., 4(2) (2009), 163–168.
[9] M. Breen, Sets of locally maximal visibility and finite unions of starshaped sets,
Monatsh. Math. 130 (2000), 1–5.
[10] J. Cell, Tietze-type theorem for partially convex planar sets, Bull. Soc. Roy. Sci.
Liege 69(1) (2000), 17–20.
[11] R.J. Dwilewicz, A short history of Convexity, Diff. Geom. Dyn. Syst. 11 (2009),
112–129.
[12] A.T. Fasto, F. Cunto, Local characterization of starshapedness, Geometriae Ded-
icata 66 (1997), 65–67.
[13] S.M. Goto , Manifolds without conjugate points, J. Diff. Geom. 13 (1978), 341–
359.
[14] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol(1), In-
terscience Publishers, New York, 1963.
[15] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol(2), In-
terscience Publishers, New York, 1969.
[16] R. Sacksteder, E.G. Straus, and F.A. Valentine, A generalization of a theorem of
Tietze and Nakajima on local convexity, Journal London Math. Soc. 36 (1961),
52–56.
[17] F.A. Valentine, Convex Sets, McGraw-Hill, Inc., New York, 1964.
[18] B. Wu On complete hypersurfaces with two distinct principal curvatures in a
hyperbolic space, Balkan J. Geom. Appl., 15(2) (2010), 126–137.
Authors’ addresses:
S. Shenawy
Basic Science Department,
Modern Academy for Engineering and Technology, Maadi, Egypt.
E-mail: s.shenawy@s-math.org , sshenawy08@yahoo.com