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 and M 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−1x (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
°Balkan Society of Geometers, Geometry Balkan Press 2008.
Definition.The hypersurfaceM is calledconvexatx∈M if there exists an open set U ⊂ V ⊂N containingx such that M ∩U is contained either in T 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 1.1If M 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)00(0) = 2((x1)0(0))2 and hence the function
f :I→R, f(t) =hc(t)−0, ξ(0)i
satisfies the relationsf(t) =x3(t) andf00(0) = (x3)00(0) = 2((x1)0(0))2.
Since Ω0(c. (0),c. (0)) = f00(0), and c is an C2 arbitrary curve, one gets that Ω0
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 point x∈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 a0 =f(x0), then for any real numbera∈Imf\{a0}, the hypersurfaceMa=f−1{a}
is strictly convex.
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−1x (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 GHx+.
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 GHx+.
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 GHx+ 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 =f0(0) =ωx(dexp−1x (c(0)))(c.(0)) =ωx(X),
(2.3) 0≤f00(0) =ωx(d2exp−1x (c(0)))(c.(0),c.(0))
+ωx(dexp−1x (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 Trace(Ωx) =g(
Xn i=1
h(ei, ei), Hx) =ng(Hx, Hx)>0,
the quadratic form Ωx cannot be negative semidefinite, therefore M ∩U cannot be contained inT GHx−.So, if the submanifoldM isH-convex at the pointx, then there exists an open setU ⊂V ⊂N containingxsuch thatM∩U is contained inT GHx+. 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.
We fixx∈M ⊂N and a neighborhoodV of xfor which expx :TxN →V is a diffeomorphism. The function
Fωx:V →R, Fωx(y) =ωx(exp−1x (y) 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, c(0) =x,c.(0) =X. The functionf =Fωx◦c:I→Rsatisfies
f0(0) =ωx(dexp−1x (c(0))(c.(0)) =ωx(c.(0)) =ωx(X) =g(H, X) = 0 and hencex∈M is a critical point ofFωx.
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= HessMFωx−dFωx(ΩH).
AsFωxis affine on each geodesic radiating fromx, it follows HessNFωx= 0.It remains that
HessMFωx(x) = Ωx
and hence HessMFωx 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.
Theorem 2.3 Let (N, g) be a Riemannian manifold and c : I → N a regular curve which have the mean curvature nonzero, whereI is an real interval. Thenc is a strictlyH-convex submanifold ofN.
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{e01, e02, ..., e0k} is an orthonormal frame inL, with e01=X, we denote
RicL(X) = Xk
j=2
k(e01∧e0j),
where k(e01∧e0j) is the sectional curvature given by Sp{e01, e0j}. We define the Ricci curvature of k-orderat the pointx∈M,
θk(x) = 1
k−1 min
L,dimL=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−1n (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 = kHHx
xk. The second fundamental form of the submanifoldM satisfies the relation
(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) R(X, Y, X, Ye ) =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∇, ∇0 and∇e the Levi-Civita connections onM,SmandRm+1 and with hthe second fundamental form ofM inRm+1, withh0 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) ∇0XY =∇XY +h0(X, Y)
and
(3.7) ∇eXY =∇0XY +eh(X, Y) =∇XY +h0(X, Y) +eh(X, Y).
Therefore
(3.8) h(X, Y) =h0(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) =h0(ei, ei) +eh(ei, ei),∀i∈1, n and hence
(3.10) H =H0+ 1 n
Xn i=1
eh(ei, ei),
where H is the mean curvature vector field of M ⊂ Rm+1 and H0 is the mean curvature vector field ofSm⊂Rm+1. We introduce the quadratic forms
Ω, Ω0 :TxM ×TxM →R,
Ω(X, Y) =hh(X, Y), Hi, Ω0(X, Y) =hh0(X, Y), H0i.
From (3.8) and (3.10), we obtain
(3.11) Ω(X, Y) =hh(X, Y), Hi=hh0(X, Y) +eh(X, Y), H0+ 1 n
Xn
i=1
eh(ei, ei)i.
Using the fact that h0(X, Y) and H0 are tangent vectors at Sm, and eh(X, Y) and Pn
i=1eh(ei, ei) are normal vectors atSm, one gets
(3.12) Ω(X, Y) =hh0(X, Y), H0i+heh(X, Y),1 n
Xn i=1
eh(ei, ei)i
= Ω0(X, Y) +heh(X, Y),1 n
Xn i=1
eh(ei, ei)i.
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.
Proof.We denote withXe the position vector field ofSm. The second fundamental form ofSm⊂Rm+1 is given by
(3.13) eh(X, Y) =heh(X, Y),Xie Xe =h∇eXY,Xie Xe
=−hY,∇eXXie Xe =−hX, YiX,e ∀X, Y ∈ X(M).
Using (3.13), we find
(3.14) 1
n Xn
i=1
eh(ei, ei) =−1 n
Xn i=1
hei, eiiXe =−X.e
From (3.12), (3.13), (3.14) and∀X, Y ∈TxM, one gets
(3.15) Ω(X, Y) = Ω0(X, Y) +hhX, YiX,e Xei= Ω0(X, Y) +hX, Yi
We read (3.15) in two ways: (i) IfM is a submanifold inSm, H-convex atx,then Ω0(x) is positive semidefinite. Using the fact that h, i is positive definite, it follows
that Ω(x) is positive definite, thereforeM is a strictlyH-convex submanifold inRm+1 atx.(ii) IfAH> In,thenhAHX, Xi>kXk2, ∀X∈TxM.Therefore
Ω0(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 Ω0 = 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: [email protected] Teodor Oprea
University of Bucharest,
Faculty of Mathematics and Informatics, Department of Geometry, 14 Academiei Str., RO-010014, Bucharest 1, Romania.
E-mail: [email protected]
