Constantin Udri¸ste and Teodor Oprea
Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined and in-vestigated the convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurface. This paper defines and studies the H-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector of the submanifold. The main results include: some prop-erties ofH-convex submanifolds, a characterization of the Chen definition of strictly H-convexity for submanifolds in real space forms [2], [3] and examples.
M.S.C. 2000: 53C40, 53C45, 52A20.
Key words: global submanifolds, convexity, mean curvature.
1
Convex hypersurfaces in Riemannian manifolds
Let (N, g) be a complete finite-dimensional Riemannian manifold andM be an ori-ented hypersurface whose induced Riemannian metric is also denoted byg.We denote byω the 1-form associated to the unit normal vector fieldξon the hypersurfaceM.
Let xbe a point in M ⊂N and V a neighborhood of x in N such that expx :
TxN →V is a diffeomorphism. The real-valued function defined onV by
F(y) =ωx(exp−x1(y))
has the property that the set
T GHx={y∈V| F(y) = 0}
is a totally geodesic hypersurface at x, tangent to M at x. This hypersurface is the common boundary of the sets
T GHx−={y∈V | F(y)≤0}, T GHx+={y∈V | F(y)≥0}.
∗
Balkan Journal of Geometry and Its Applications, Vol.13, No.2, 2008, pp. 112-119. c
Definition.The hypersurfaceM is calledconvexatx∈M if there exists an open set U ⊂ V ⊂N containingx such that M ∩U is contained either inT GHx− or in
T GHx+.
A hypersurfaceM convex atxis said to be strictly convexatxif
M ∩U∩T GHx={x}.
In [7] it was obtained a necessary condition for a hypersurface of a Riemannian manifold to be convex at a given point.
Theorem 1.1IfM is an oriented hypersurface inN, convex atx∈M, then the bilinear form
Ωx:TxM×TxM →R, Ωx(X, Y) =g(h(X, Y), ξ),
where ξ is the normal versor at x, and h is the second fundamental form of M, is semidefinite.
The converse of Theorem 1.1 is not true. To show this, we consider the surface
M :x3= (x1)2+ (x2)3in R3. One observes that 0∈M,ξ(0) = (0,0,1) andT GH0: x3= 0 is the plane tangent toM at the origin. On the other hand, if
c:I→M, c(t) = (x1(t), x2(t), x3(t))
is aC2 curve such that 0∈I andc(0) = 0, then (x3)′′(0) = 2((x1)′(0))2 and hence
the function
f :I→R, f(t) =hc(t)−0, ξ(0)i
satisfies the relationsf(t) =x3(t) andf′′(0) = (x3)′′(0) = 2((x1)′(0))2.
Since Ω0(
.
c (0),c. (0)) = f′′(0), and c is an C2 arbitrary curve, one gets that Ω 0
is positive semidefinite. HoweverM is not convex at the origin because the tangent planeT GH0:x3= 0 cuts the surface along the semicubic parabola
x3= 0,(x1)2+ (x2)3= 0
and consequently in any neighborhood of the origin there exist points of the surface placed both below the tangent plane and above the tangent plane.
If the bilinear form Ω is definite at the pointx∈M, then the hypersurfaceM is strictly convex atx.
The next results [7] establish a connection between the Riemannian manifolds admitting a function whose Hessian is positive definite and their convex hypersurfaces.
Theorem 1.2Suppose that the Riemannian manifold (N, g)supports a function
f :N →R with positive definite Hessian. On each compact oriented hypersurfaceM
inN there exists a pointx∈M such that the bilinear formΩ(x)is definite.
Theorem 1.3If the Riemannian manifold(N, g)supports a function f :N →R
with positive definite Hessian, then
1) there is no compact minimal hypersurface inN;
2) if the hypersurface M is connected and compact and its Gauss curvature is nowhere zero, thenM is strictly convex.
Theorem 1.4Let (N, g)be a connected and complete Riemannian manifold and
f :N →Ra function with positive definite Hessian. If x0 is a critical point of f and
2
H-convex Riemannian submanifolds
Having in mind the model of convex hypersurfaces in Riemannian manifolds, we define theH-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector of the submanifold.
Let (N, g) be a complete finite-dimensional Riemannian manifold and M be a submanifold inN of dimension nwhose induced Riemannian metric is also denoted byg.Letxbe a point inM ⊂N, withHx6= 0 andV a neighborhood ofxinN such that expx:TxN →V is a diffeomorphism. We denote by ω the 1-form associated to themean curvature vectorH ofM.
The real-valued function defined onV by
F(y) =ωx(exp−x1(y))
has the property that the set
T GHx={y∈V| F(y) = 0}
is a totally geodesic hypersurface at x, tangent to Mat x. This hypersurface is the common boundary of the sets
T GHx− ={y∈V| F(y)≤0}, T GHx+={y ∈V| F(y)≥0}.
Definition. The submanifold M is called H-convex at x∈M if there exists an open setU ⊂V ⊂N containingxsuch thatM∩U is contained either inT GHx− or inT GH+
x.
A submanifoldM, which isH-convex atx, is calledstrictly H-convexat xif
M ∩U∩T GHx={x}.
The next result is a necessary condition for a submanifold of a Riemannian man-ifold to beH-convex at a given point.
Theorem 2.1If M is a submanifold inN, H-convex atx∈M, then the bilinear form
Ωx:TxM×TxM →R,Ωx(X, Y) =g(h(X, Y), H), wherehis the second fundamental form ofM, is positive semidefinite.
Proof. We suppose that there is a open setU ⊂V ⊂N which contains the point
xsuch thatM ∩U ⊂T GH+
x.
For an arbitrary vectorX ∈TxM,letc:I→M ∩U be aC2curve, where I is a real interval such that 0∈I andc(0) =x,c.(0) =X.Asc(I)⊂M∩U ⊂T GH+
x the functionf =F◦c:I→Rsatisfies
(2.1) f(t)≥0, ∀t∈I.
It follows that 0 is a global minimum point forf, and hence
(2.2) 0 =f′(0) =ωx(dexp−x1(c(0)))( .
c(0)) =ωx(X),
(2.3) 0≤f′′(0) =ωx(d2exp−x1(c(0)))( .
+ωx(dexp−x1(c(0)))( ..
c(0)) =ωx( ..
c(0)) = Ωx(X, X).
SinceX∈TxM is an arbitrary vector, we obtain that Ωxis positive semidefinite.
Remark.We consider{e1, e2, ..., en}an orthonormal frame inTxM. Since
the quadratic form Ωx cannot be negative semidefinite, therefore M ∩U cannot be contained inT GH−
x.So, if the submanifoldM isH-convex at the pointx, then there exists an open setU ⊂V ⊂N containingxsuch thatM∩U is contained inT GH+
x. In the sequel, we prove that if the bilinear form Ωx is positive definite, then the submanifoldM is strictlyH-convex at the point x. For this purpose we introduce a function similar to the height function used in the study of the hypersurfaces of an Euclidean space.
has the property that it is affine on geodesics radiating fromx.
We consider an arbitrary vectorX ∈TxM and a curvec:I→V such that 0∈I,
Theorem 2.2Let M be a submanifold inN. If the bilinear form Ωx is positive definite, thenM is strictlyH-convex at the pointx.
Proof. The point x∈M is a critical point of Fωx andFωx(x) = 0. On the other
hand one observes that
HessNFωx= Hess
M
Fωx−dFωx(ΩH).
AsFωxis affine on each geodesic radiating fromx, it follows Hess
NF
ωx is positive definite at the pointx. In this wayxis a strict local
minimum point forFωx inM∩V, i.e., the submanifold M is strictlyH-convex at x.
Remark. 1) The bilinear form Ωx is positive (semi)definite if and only if the Weingarten operatorAH is positive (semi)definite.
2) If M is an hypersurface in N, x is a point in M with Hx 6= 0, then M is
H-convex atxif and only ifM is convex at x.
A class of strictly H-convex submanifolds into a Riemannian manifold is made from the curves which have the mean curvature nonzero.
Proof.We fixt∈I. AsTc(t)c=Sp{
.
c(t)}, we obtain
Hc(t)=
h(c.(t),c.(t)) °
°.
c(t)°°2 .
Since Ω(c. (t),c. (t)) =g(h(c. (t),c. (t)), Hc(t)) =
°
°c. (t)°°2°°Hc(t)
° °2
>0, the quadratic form Ω is positive definite. It follows that the curvecis a strictlyH-convex subman-ifold ofN.
3
H-convex Riemannian submanifolds
in real space forms
Let us consider (M, g) a Riemannian manifold of dimension n. We fix x ∈ M and
k∈2, n.LetLbe a vector subspace of dimensionkinTxM. IfX ∈Lis a unit vector, and{e′
1, e′2, ..., e′k} is an orthonormal frame inL, with e′1=X, we denote
RicL(X) = k
X
j=2
k(e′1∧e′j),
where k(e′1∧e′j) is the sectional curvature given by Sp{e′1, e′j}. We define the Ricci curvature of k-orderat the pointx∈M,
θk(x) = 1
k−1 L,dimminL=k,
X ∈L,kXk= 1
RicL(X).
B. Y. Chen showed [2], [3] that the eigenvalues of the Weingarten operator of a submanifold in a real space form and the Ricci curvature ofk-order satisfies the next inequality.
Theorem 3.1Let (Mf(c),eg)be a real space form of dimensionmandM ⊂Mf(c) a submanifold of dimensionn, andk∈2, n. Then
(i)AH ≥n−n1(θk(x)−c)In.
(ii) Ifθk(x)6=c,then the previous inequality is strict.
Corollary 3.2If M is a submanifold of dimensionnin the real space formMf(c) of dimension m,x∈M and there is a natural numberk∈2, n such that θk(x)> c, thenM is strictly H-convex at the pointx.
The converse of previous corollary is also true in the case of hypersurfaces in a real space form.
Theorem 3.3 If M is a hypersurface of dimensionn of a real space form Mf(c) andM is strictlyH-convex at a pointx, then
θk(x)> c, ∀k∈2, n.
Proof.Letxbe a point inM,letH be the mean curvature of M andπa 2-plane inTxM. We consider{X, Y} an orthonormal frame inπ andξ = Hx
kHxk. The second
(3.1) h(U, V) = Ωx(U, V)
kHxk
ξ, ∀U, V ∈TxM.
On the other hand, the Gauss equation can be written
(3.2) Re(X, Y, X, Y) =R(X, Y, X, Y)−eg(h(X, X), h(Y, Y)) +eg(h(X, Y), h(X, Y)).
Using the relation (3.1) and the fact thatMf(c) has the sectional curvaturec, we obtain
(3.3) R(X, Y, X, Y) =c+ 1
kHxk2
(Ωx(X, X)Ωx(Y, Y)−Ωx(X, Y)2).
On the other hand, Ωx is positive definite because M is strictly H-convex at the point x. From the Cauchy inequality, using the fact that X and Y are linear independent vectors, it follows
(3.4) Ωx(X, X)Ωx(Y, Y)−Ω(X, Y)2>0.
From (3.3) and (3.4) we find
(3.5) R(X, Y, X, Y)> c,
which means that the sectional curvature ofM at the pointxis strictly greater than
c.Using the definition of Ricci curvatures, it follows that
θk(x)> c, ∀k∈2, n.
LetM be a submanifold of dimensionnin themdimensional sphereSm⊂Rm+1.
We denote withh , i the metrics induced on Sm and M by the standard metric of
Rm+1, with∇, ∇′ and∇e the Levi-Civita connections onM,SmandRm+1 and with hthe second fundamental form ofM inRm+1, withh′
the second fundamental form ofM inSmand withehthe second fundamental form ofSmin Rm+1.
LetX, Y be two vector fields tangents toM. The Gauss formula gives
(3.6) ∇′XY =∇XY +h
′ (X, Y)
and
(3.7) ∇eXY =∇ ′
XY +eh(X, Y) =∇XY +h ′
(X, Y) +eh(X, Y).
Therefore
(3.8) h(X, Y) =h′(X, Y) +eh(X, Y).
We fix a pointx∈M and an orthonormal frame{e1, e2, ..., en}inTxM.From the relation (3.8), one gets
(3.9) h(ei, ei) =h ′
(ei, ei) +eh(ei, ei),∀i∈1, n
(3.10) H =H′+ 1 curvature vector field ofSm⊂Rm+1. We introduce the quadratic forms
Ω, Ω′ :TxM ×TxM →R,
Ω(X, Y) =hh(X, Y), Hi, Ω′(X, Y) =hh′(X, Y), H′i.
From (3.8) and (3.10), we obtain
(3.11) Ω(X, Y) =hh(X, Y), Hi=hh′(X, Y) +eh(X, Y), H′+ 1
Based on these considerations, we formulate the next
Theorem 3.4We consider a pointx∈M.
(i) IfM is a submanifold in Sm, H-convex at x, thenM is strictly H-convex at
x, as submanifold inRm+1.
(ii) If the Weingarten operatorAH ofM ⊂Rm+1satisfies the inequalityAH> In, thenM is a submanifold inSm,strictly H-convex at x.
that Ω(x) is positive definite, thereforeM is a strictlyH-convex submanifold inRm+1
atx.(ii) IfAH> In,thenhAHX, Xi>kXk2, ∀X∈TxM.Therefore
Ω′(X, X) = Ω(X, X)− kXk2
=hh(X, X), Hi − kXk2
=hAHX, Xi − kXk2>0,∀X ∈TxM. ConsequentlyM is a submanifold in Sm,strictlyH-convex atx.
Corollary 3.5IfM is a minimal submanifold inSm,thenM is strictlyH-convex atxas submanifold in Rm+1.
Proof. Using the fact that M is minimal in Sm, one gets Ω′ = 0, therefore Ω(X, Y) =hX, Yi,∀X, Y ∈ X(M).Consequently Ω is positive definite.
References
[1] R.L. Bishop,Infinitesimal convexity implies local convexity, Indiana Univ. Math. J. 24, 2 (1974), 169-172.
[2] B.Y. Chen,Mean curvature shape operator of isometric immersions in real-space-forms, Glasgow Math. J. 38 (1996), 87-97.
[3] B.Y. Chen, Relations between Ricci curvature shape operator for submanifolds with arbitrary codimensions,Glasgow Math. J. 41(1999), 33-41.
[4] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol 1,2, Inter-science, New York, 1963, 1969.
[5] J.A. Thorpe,Elementary topics in differential geometry, Springer-Verlag 1979. [6] C. Udri¸ste,Convex hypersuprafaces, Analele S¸t. Univ. Al. I. Cuza, Ia¸si, 32 (1986),
85-87.
[7] C. Udri¸ste,Convex Functions and Optimization Methods on Riemannian Man-ifolds, Mathematics and Its Applications, 297, Kluwer Academic Publishers Group, Dordrecht, 1994.
Authors’ addresses:
Constantin Udri¸ste
Department of Mathematics-Informatics I, Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independent¸ei,
RO-060042 Bucharest, Romania. E-mail: udriste@mathem.pub.ro
Teodor Oprea
University of Bucharest,
Faculty of Mathematics and Informatics, Department of Geometry, 14 Academiei Str., RO-010014, Bucharest 1, Romania.
