Wintgen ideal submanifolds with a low-dimensional integrable distribution (I)

0.17 MB·English

by  Tongzhu Li

Wintgen ideal submanifolds with a
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1
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low-dimensional integrable distribution (I)
2

n
a
J
Tongzhu Li, Xiang Ma, Changping Wang

2
2

]
G
D Abstract
.
h
Asubmanifoldinspaceformssatisfiesthe well-knownDDVVinequality dueto
t
a
m De Smet, Dillen, Verstraelen and Vrancken. The submanifold attaining equality
[ in the DDVV inequality at every point is called Wintgen ideal submanifold. As

2 conformal invariant objects, Wintgen ideal submanifolds are studied in this paper
v
using the framework of Mo¨bius geometry. We classify Wintgen ideal submanfi-
2
4
olds of dimension m > 2 and arbitrary codimension when a canonically defined
7
4 2-dimensionaldistribution D is integrable. Such examples come from cones, cylin-
.
1 ders,orrotationalsubmanifolds oversuper-minimalsurfacesin spheres,Euclidean
0
3 spaces, or hyperbolic spaces, respectively.
1
:
v
i 2000 Mathematics Subject Classification: 53A30,53A55;
X
r Key words: Wintgen ideal submanifolds,DDVV inequality, super-conformalsurfaces,
a
super-minimal surfaces.
1 Introduction
A basic idea in submanifold theory is to findcertain universal inequalities (pointwise
or global ones) between various invariants (intrinsic and extrinsic), then characterize
and classify the optimal submanifolds attaining equality in such inequalities.
0T. Li, X. Ma, C.P. Wang are partially supported by the grant No. 11171004 of NSFC; X. Ma is
partially supported by thegrant No. 10901006 of NSFC.
1
The simplest example of such pointwise inequalities is that for mean curvature H
and Gaussian curvature K of a surface in R3, there is always H2 K, and the equality
≥
holds true exactly at those umbilic points. This was generalized by Chen [3] to other
space forms and to arbitrary codimensional case. (Another universal inequality posed
by Chen [2] motivated a series of investigation on submanifolds attaining the equality
in Chen’s inequality.)
De Smet, Dillen, Verstraelen and Vrancken proposedin 1999 a strengthened inequal-
ity [7]involving thescalar, mean curvatureand the norm of th normal curvaturetensor
as below. Let f : Mm Qm+p be an isometric immersion of an m dimensional
c
−→ −
Riemannian manifold into a space form of dimension m + p and constant sectional
curvature c. Let R (resp. R⊥) be the Riemannian curvature tensor (resp. the normal
curvature tensor) of f. Their conjectured that at any point,
(1.1) DDVV inequality : s c+ H 2 s⊥.
≤ || || −
Here H denotes the mean curvature of f, and
2 2
s = R(e ,e )e ,e , s⊥ = R⊥ .
i j j i
m(m 1) h i m(m 1)|| ||
− 1≤i<j≤n −
X
This so-called DDVV conjecture was proved in 2008 by J. Ge, Z. Tang [10] and Z. Lu
[14] independently.
Moreover, in [10] the pointwise structure of the second fundamental form of f that
attains equality was determined. It was shown that equality holds at x Mm if, and
∈
only if, there exists an orthonormal basis e , ,e in the tangent space T Mm and
1 m x
{ ··· }
an orthonormal basis n , ,n in the normal space T⊥Mm, such that the shape
1 p x
{ ··· }
operators A ,i = 1, ,m have the form
{ ni ··· }
λ µ 0 0 λ +µ 0 0 0
1 0 2 0
··· ···
   
µ λ 0 0 0 λ µ 0 0
0 1 2 0
··· − ···
(1.2) A = 0 0 λ 0,A =  0 0 λ 0,
n1  ... ... ...1 ·.·.·. ...  n2  ... ... ...2 ·.·.·. ... 
   
   
0 0 0 λ 0 0 0 λ
 1  2
 ···   ··· 
   
and
A = λ I , A = 0,r 4.
n3 3 p nr ≥
2
Wintgen first proved the inequality (1.1) for surfaces f : M2 S4. The equality
→
holds at x M2 if and only if the curvature ellipse of M2 in S4 at x is a circle [18, 11].
∈
f : M2 S4 is called a super-conformal surface if this holds true at every point. It is
→
well-known that such surfaces correspond to images of complex curves in CP3 via the
Penrose twistor projection π : CP3 S4. According to the suggestion of [4, 16], we
−→
make the following definition.
Definition 1.1. A submanifold f : Mm Qm+p ia called a Wintgen ideal subman-
c
−→
ifold if the equality is satisfied in (1.1) at every point of Mm.
We briefly review known results on the classification of Wintgen ideal submanifolds.
It was shown in [11] that f : M2 Q2+p is a Wintgen ideal surface if and only if
c
−→
the ellipse of curvature of f at x is a circle. When being also minimal surfaces in the
specificspaceform,theywerealreadyknownassuper-minimal surfaces. Suchexamples
are abundant.
ForthreedimensionalWintgenidealsubanifolds,whentheyareminimal,theybelong
totheclass of(threedimensional)austeresubmanifolds. Thelater was classifiedlocally
by Bryant for Euclidean submanifolds [1], by Dajczer and Florit in the unit sphere [5],
by Choi and Lu [13, 15] in hyperbolic space.
Finally, in [6], Dajczer and Tojeiro provided a parametric construction of Wintgen
ideal submanifolds of codimension two and arbitrary dimension in terms of minimal
surfaces in the Euclidean space.
An important observation of Dajczer and Tojeiro in [6] is that the inequality as
well as the equality case (and the class of Wintgen ideal submanifolds) are conformally
invariant property. This follows from the observation in [8] that inequality (1.1) holds
at a point x Mm if and only if
∈
p p 2
[A¯ ,A¯ ] 2 A¯
α β α
|| || ≤ || ||
!
α,β=1 α=1
X X
is satisfied for the traceless shape operators A¯ , ,A¯ at x Mm, whereas A¯
1 p i
··· ∈ { }
are conformal invariant objects (up to a scalar factor). It follows that Wintgen ideal
submanifolds in the sphere Sm+p or hyperbolic space Hm+p are the pre-image of a
stereographic projection of Wintgen ideal submanifolds in Rm+p.
3
Thus it is appropriate to put the study of Wintgen ideal submanifolds in the frame-
work of M¨obius geometry. For the same reason it is no restriction when we describe
them in the Euclidean space. This is exactly our main goal in this paper.
As the main result, we give a classification of Wintgen ideal submanifolds with
integrable canonical distribution D = Span e ,e . It is clear from (1.2) that D is
1 2
{ }
well-defined when the submanifold is nowhere totally umbilic.
Theorem 1.1. Let f : Mm Rm+p(m 3) be a Wintgen ideal submanifold without
−→ ≥
umbilic points. If the canonical distribution D = span e ,e is integrable, then locally
1 2
{ }
f is Mo¨bius equivalent to
(i) a cone over a super-minimal surface in S2+p;
(ii) or a cylinder over a super-minimal surface in R2+p;
(iii) or a rotational submanifold over a super-minimal surface in H2+p.
Thispaperisorganized asfollows. Insection 2, wereviewtheelementary facts about
M¨obius geometry for submanifolds in Rm+p. In section 3, we describe the construc-
tion of Wintgen ideal submanifolds as cylinders, cones, or rotational submanifolds. In
section 4, we give the proof of our main theorem.
2 Submanifold theory in M¨obius geometry
In this section we briefly review the theory of submanifolds in M¨obius geometry. For
details we refer to [17] and [12].
Let Rm+p+2 be the Lorentz space with inner product , defined by
1 h· ·i
Y,Z = Y Z +Y Z + +Y Z ,
0 0 1 1 m+p+1 m+p+1
h i − ···
where Y = (Y ,Y , ,Y ),Z = (Z ,Z , ,Z ) Rm+p+2.
0 1 m+p+1 0 1 m+p+1
··· ··· ∈
Let f : Mm Rm+p be a submanifold without umbilics and assume that e is an
i
→ { }
orthonormal basis with respect to the induced metric I = df df with θ the dual
i
· { }
4
basis. Let n 1 r p bea local orthonormal basis for the normal bundle. As usual
r
{ | ≤ ≤ }
we denote the second fundamental form and the mean curvature of f as
1
II = hr θ θ n , H = hr n = Hrn .
ij i⊗ j r m jj r r
ij,γ j,r r
X X X
We define the M¨obius position vector Y :Mm Rm+p+2 of f by
→ 1
1+ f 2 1 f 2 m 1 2
Y = ρ | | , −| | ,f , ρ2 = II tr(II)I .
2 2 m 1 − m
(cid:18) (cid:19) − (cid:12) (cid:12)
(cid:12) (cid:12)
It is known that Y is a well-defined canonical lift of f. T(cid:12)(cid:12)wo submanifolds(cid:12)(cid:12) f,f¯: Mm
→
Rm+p are M¨obius equivalent if there exists T in the Lorentz group O(m+p+1,1) in
Rm+p+2 such that Y¯ = YT. It follows immediately that
1
g = dY,dY = ρ2dx dx
h i ·
is a M¨obius invariant, called the M¨obius metric of f.
Let ∆ be the Laplacian with respect to g. Define
1 1
N = ∆Y ∆Y,∆Y Y,
−m − 2m2h i
which satisfies
Y,Y = 0= N,N , N,Y = 1 .
h i h i h i
Let E , ,E bealocalorthonormalbasisfor(Mm,g)withdualbasis ω , ,ω .
1 m 1 m
{ ··· } { ··· }
Write Y = E (Y). Then we have
j j
Y ,Y = Y ,N = 0, Y ,Y = δ , 1 j,k m.
j j j k jk
h i h i h i ≤ ≤
We define
1+ f 2 1 f 2
ξ = Hr | | , −| | ,f +(f n , f n ,n ).
r r r r
2 2 · − ·
(cid:18) (cid:19)
Then ξ , ,ξ betheorthonormalbasisoftheorthogonalcomplementofSpan Y,N,Y 1
1 p j
{ ··· } { | ≤
j m . And Y,N,Y ,ξ form a moving frame in Rm+p+2 along Mm.
≤ } { j r} 1
Remark 2.1. Geometrically, ξ corresponds to the unique sphere tangent to M at
r m
one point x with normal vector n and the same mean curvature Hr(x). We call ξ
r r
{ }
the mean curvature spheres of Mm.
5
We willusethefollowing rangeof indicesin thissection: 1 i,j,k m;1 r,s p.
≤ ≤ ≤ ≤
We can write the structure equations as below:
dY = ω Y ,
i i
i
X
γ
dN = A ω Y + C ω ξ ,
ij i j i i r
ij i,γ
X X
γ
dY = A ω Y ω N + ω Y + B ω ξ ,
i − ij j − i ij j ij j r
j j j,γ
X X X
dξ = Crω Y ω BrY + θ ξ ,
r i i i ij j rs s
− −
i ijr s
X X X
where ω are the connection 1-forms of the M¨obius metric g and θ the normal con-
ij rs
nection 1-forms. The tensors
A = A ω ω , B = Brω ω ξ , Φ = Crω ξ
ij i j ij i j r j j r
⊗ ⊗
ij ijr jr
X X X
are called the Blaschke tensor, the M¨obius second fundamental form and the M¨obius
form of x, respectively. The covariant derivatives of Cr,A ,Br are defined by
i ij ij
Cr ω =dCr + Crω + Csθ ,
i,j j i j ji j sr
j j s
X X X
A ω = dA + A ω + A ω ,
ij,k k ij ik kj kj ki
k k k
X X X
Br ω = dBr + Br ω + Br ω + Bsθ .
ij,k k ij ik kj kj ki ij sr
k k k s
X X X X
The integrability conditions for the structure equations are given by
(2.3) A A = Br Cr BrCr,
ij,k ik,j ik j ij k
− −
r
X
(2.4) Cr Cr = (Br A Br A ),
i,j j,i ik kj jk ki
− −
k
X
(2.5) Br Br = δ Cr δ Cr,
ij,k ik,j ij k ik j
− −
(2.6) R = Br Br BrBr +δ A +δ A δ A δ A ,
ijkl ik jl il jk ik jl jl ik il jk jk il
− − −
r
X
(2.7) R⊥ = Br Bs Bs Br .
rskl ik kj − ik kj
k
X
Here R denote the curvature tensor of g, κ = 1 R is its normalized
ijkl n(n−1) ij ijij
M¨obius scalar curvature. It follows from (2.6) that the RiccPi tensor of g satisfies
(2.8) R := R = Br Br +(trA)δ +(m 2)A .
ij ikjk − ik kj ij − ij
k kr
X X
6
Other restrictions on tensors A,B are
m 1
(2.9) Br = 0, (Br)2 = − ,
jj ij m
j ijr
X X
1
(2.10) trA= A = (1+m2κ).
jj
2m
j
X
We know that all coefficients in the structure equations are determined by g,B and
{ }
the normal connection θ .
αβ
{ }
3 Examples of Wintgen ideal submanifolds
Inthis section we willuseminimalWintgen ideal submanifoldsin space formsRn,Sn
or Hn to construct other general examples. Note that being a minimal submanifold is
not a conformal invariant property.
WeremarkthataminimalWintgenidealsubmanifoldinanyspaceformisanaustere
submanifoldof ranktwo, andtheconverse is alsotrue. Super-minimalsurfaces inspace
formsarespecialexamplesandthereareplentyofthem,includingallminimal2-spheres
in Sn and all complex curves in Cn = R2n.
Definition 3.1. Let u : Mr Rr+p be an immersed submanifold. We define the
−→
cylinder over u in Rm+p as
f = (u,id) :Mr Rm−r Rr+p Rm−r = Rm+p,
× −→ ×
where id :Rm−r Rm−r is the identity map.
−→
Proposition 3.1. Let u : Mr Rr+p be an immersed submanifold. Then the
−→
cylinder f = (u,id) : Mr Rm−r Rm+p is a Wintgen ideal submanifold if and only
× −→
if u:Mr Rr+p is a minimal Wintgen ideal submanifold.
−→
Proof. Letη , ,η beanorthonormalframeinthenormalbundleofuinRr+p. Then
1 p
···
e = (η ,~0) Rm+p is such a frame of f. The first and second fundamental forms I,II
r r
∈
of f are related with corresponding forms I ,II of u by
u u
(3.11) I = Iu+IRm−r, II = IIu,
7
where IRm−r denotes the standard metric of Rm−r. Clearly f is a Wintgen ideal sub-
manifold if and only if u is a minimal Wintgen ideal submanifold. This completes the
proof to Proposition 3.1.
Remark 3.1. The Mo¨bius position vector Y :Mr Rm−r Rm+p+2 of the cylinder
× −→ 1
f is
1+ u2+ y 2 1 u2 y 2
Y = ρ | | | | , −| | −| | ,u,y ,
0
2 2
(3.12) (cid:18) (cid:19)
Y : Mr Rm−r Rr+p+2 Rm−r,
× −→ 1 ×
where ρ = m (II 2 mH2) : Mr R, and y : Rm−r Rm−r is the identity
0 m−1 | u| − u −→ −→
map.
Definition 3.2. Let u : Mr Sr+p Rr+p+1 be an immersed submanifold. We
−→ ⊂
define the cone over u in Rm+p as
f :R+ Rm−r−1 Mr Rm+p,
× × −→
f(t,y,u)= (y,tu),
Proposition 3.2. Let u : Mr Sr+p be an immersed submanifold. Then the cone
−→
f = (y,tu) :R+ Rm−r−1 Mr Rm+p is a Wintgen ideal submanifold if and only
× × −→
if u is a minimal Wintgen ideal submanifold in Sr+p.
Proof. The first and second fundamental forms of f are, respectively,
I = t2Iu+IRm−r, II = t IIu,
where Iu,IIu,IRm−r are understood as before. The conclusion follows easily.
Remark 3.2. The Mo¨bius position vector Y :R+ Rm−r−1 Mr Rm+p+2 of the
× × −→ 1
cone f is
1+t2+ y 2 1 t2 y 2
Y = ρ | | , − −| | ,y,u ,
0
2t 2t
(cid:18) (cid:19)
where ρ = m (II 2 mH2) :Mr R, and y :Rm−r−1 Rm−r−1 is the identity
0 m−1 | u| − u −→ −→
map. Let
Hm−r = (y ,y) Rm−r+1 y2+ y 2 = 1,y 1 = R+ Rm−r−1,
{ 0 ∈ |− 0 | | − 0 ≥ } ∼ ×
8
then (1+t2+|y|2,1−t2−|y|2,y) : R+ Rm−r−1 = Hm−r Hm−r is nothing else but the
2t 2t × →
identity map. And the Mo¨bius position vector of the cone f is
(3.13) Y = ρ (id,u) :Hm−r Mr Hm−r Sr+p Rm+p+2,
0 × → × ⊂ 1
where ρ C∞(Mr) and id :Hm−r Hm−r is a identity map.
0
∈ →
Definition 3.3. Let Rr+p = (x , ,x ) Rr+p x > 0 be the upper half-space
+ { 1 ··· r+p ∈ | r+p }
endowed with the standard hyperbolic metric
r+p
1
ds2 = dx2 .
x2 i
r+p i=1
X
Let u = (x , ,x ) : Mr Rr+p be an immersed submanifold. We define rota-
1 ··· r+p −→ +
tional submanifold over u in Rm+p as
f : Mr Sm−r Rm+p,
× −→
f(u,φ)= (x , ,x ,x φ).
1 r+p−1 r+p
···
where φ: Sm−r Rm−r+1 is the standard sphere.
−→
Proposition 3.3. Let u = (x , ,x ) : Mr Rr+p be an immersed subman-
1 ··· r+p −→ +
ifold. Then the rotational submanifold f : Mr Sm−r Rm+p is a Wintgen ideal
× −→
submanifold if and only if u is a minimal Wintgen ideal submanifold.
Proof. Let Rr+p+1 be the Lorentz space with inner product
1
y,y = y2+y2+ +y2 , y = (y , ,y ).
h i − 1 2 ··· r+p+1 1 ··· r+p+1
Let Hr+p = y Rr+p+1 y,y = 1,y > 0 be the hyperbolic space. Introduce
{ ∈ 1 |h i − 1 }
isometry τ :Rr+p Hr+p as below:
+ −→
(3.14)
1+x2+ +x2 1 x2 x2 x x
τ(x , ,x )= 1 ··· r+p, − 1−···− r+p, 1 , , r+p−1 .
1 r+p
··· 2x 2x x ··· x
r+p r+p r+p r+p !
Let η1, ,ηp be the unit normal vectors of u in Rr+p. Write ηi = (ηi, ,ηi ).
··· + 1 ··· r+p
Since ηi is the unit normal vector, then
(ηi)2+ +(ηi )2
1 ··· r+p = 1, 1 i p.
x2 ≤ ≤
r+p
9
Thus the unit normal vector of f in Rm+p is
1
ξ = (ηi, ,ηi φ).
i x 1 ··· r+p
r+p
The first fundamental form of u is
1
I = (dx dx + +dx dx ).
u x2 1· 1 ··· r+p· r+p
r+p
The second fundamental form of u is
1 ηi
IIi = τ (du),τ (dηi) = (dx dηi + +dx dηi ) r+pI .
u −h ∗ ∗ i x2 1· 1 ··· r+p· r+p − x u
r+p r+p
Now we can write out the first and the second fundamental forms of f:
I = dx·dx = x2r+p(Iu+ISm−r), IIi = xr+pIIui −ηri+p(Iu +Ism−r),
where ISm−r is the standard metric of Sm−r. This completes the proof.
Remark 3.3. TheMo¨biusposition vectorY : Mr Sm−r Rm+p+2 of therotational
× −→ 1
submanifold f is
1+ u2 1 u2 x x
1 r+p−1
Y = ρ ( | | , −| | , , , ,φ),
0
2x 2x x ··· x
r+p r+p r+p r+p
where ρ = m (II 2 mH2) : Mr R, and φ : Sm−r Sm−r is the identity
0 m−1 | u| − u −→ −→
map. Since (1+|u|2,1−|u|2, x1 , ,xr+p−1) = τ(u) : Mr Hr+p, then the Mo¨bius
2xr+p 2xr+p xr+p ··· xr+p →
position vector of the rotational submanifold f is
(3.15) Y = ρ (τ(u),φ) :Mr Sm−r Hr+p Sm−r Rm+p+2,
0 × → × ⊂ 1
where ρ C∞(Mr) and φ: Hm−r Hm−r is the identity map.
0
∈ →
From (3.12), (3.13) and (3.15), we have
Proposition 3.4. Letf : Mm Rm+p be animmersed submanifold without umbilical
→
points.
(1) If there exists a submanifold u : Mr Rr+p such that the Mo¨bius position vector
→
of f is
1+ u2 + y 2 1 u2 y 2
Y = ρ | | | | , −| | −| | ,u,y
0
2 2
(cid:18) (cid:19)
Y : Mr Rm−r Rr+p+2 Rm−r Rm+p+2,
× −→ 1 × ⊂ 1
10