C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIII 1992 FASC. 2

VOL. LXIII 1992 FASC. 2

QUADRIC HYPERSURFACES OF FINITE TYPE

BY

BANG - YEN C H E N (EAST LANSING, MICHIGAN), FRANKI D I L L E N † (LEUVEN)

AND

HONG - ZAO S O N G ‡ (HENAN)

Introduction. A submanifold M of the Euclidean m-space E ^m is said to be of finite type (see [C1] for details) if each component of its position vector field X can be written as a finite sum of eigenfunctions of the Laplacian ∆ of M , i.e., if

X = X 0 + X 1 + . . . + X k

where X 0 is a constant vector and ∆X t = λ t X t for t = 1, . . . , k. If in particular all eigenvalues λ 1 , . . . , λ k are mutually different, then M is said to be of k-type. If we define a polynomial P by

P (T ) =

k

Y

t=1

(T − λ t ) ,

then P (∆)(X − X 0 ) = 0. If M is compact, then the converse also holds, i.e., if there exists a constant vector X 0 and a nontrivial polynomial P such that P (∆)(X − X 0 ) = 0, then M is of finite type [C1].

The class of finite type submanifolds is very large, including minimal submanifolds of E ^m , minimal submanifolds of a hypersphere, parallel sub- manifolds, compact homogeneous submanifolds equivariantly immersed in a Euclidean space, and also isoparametric hypersurfaces of a hypersphere.

On the other hand, very few hypersurfaces of finite type in a Euclidean space are known, other than minimal hypersurfaces (which are of 1-type).

Therefore the following problem seems to be quite interesting.

Problem. Classify all finite type hypersurfaces in E ^m .

For m = 2, this problem was solved completely. In fact, it is known that

† Supported by a research fellowship of the Research Council of the Katholieke Uni- versiteit Leuven.

‡ This work was done while the third author was a visiting scholar at Michigan State

University. He would like to take this opportunity to express his hearty thanks to his

colleagues there for their hospitality.

circles and straight lines are the only curves of finite type in E ² (see [C1]

and [CDVV] for details). For m = 3, the first result in this respect given in [C2], states that circular cylinders are the only tubes in E ³ which are of finite type. In [CDVV] it is shown that a ruled surface in E ³ is of finite type if and only if it is a plane, a circular cylinder or a helicoid. In [G], it is shown that a cone in E ^m is of finite type if and only if it is minimal. In [D], some ruled submanifolds of finite type are classified.

If M ⁰ is an algebraic hypersurface with singularities in E ⁿ , then M ⁰ is said to be of finite type if M ⁰ − {singularities} is of finite type.

Combining the notion of algebraic hypersurfaces and the notion of sub- manifolds of finite type, the first two authors proved in [CD] that the only quadric surfaces of finite type in E ³ are the circular cylinders and the spheres. In this article, we shall completely classify quadric hypersurfaces of finite type.

2. Quadric hypersurfaces. A subset M of an n-dimensional Euclidean space E ⁿ is called a quadric hypersurface if it is the set of points (x 1 , . . . , x n ) satisfying the following equation of the second degree:

(2.1)

n

X

i,k=1

a ik x i x k +

n

X

i=1

b i x i + c = 0 ,

where a ik , b i , c are all real numbers. We can assume without loss of gen- erality that the matrix A = (a ik ) is symmetric and A is not a zero matrix.

By applying a coordinate transformation in E ⁿ if necessary, we may assume that (2.1) takes one of the following canonical forms:

(I)

r

X

i=1

a i x ² _i + 1 = 0 ,

(II)

r

X

i=1

a i x ² _i + 2x r+1 = 0 ,

(III)

r

X

i=1

a i x ² _i = 0

where (a 1 , . . . , a r , 0, . . . , 0) (with n − r zeros) is proportional to the eigen-

values of the matrix A. In general, we have 1 ≤ r ≤ n. In the cases where

r = n in (I) and (III) and r + 1 = n in (II) the hypersurface is called a prop-

erly (n − 1)-dimensional quadric hypersurface, and in other cases, a quadric

cylindrical hypersurface. In cases (I) and (III), the quadric cylindrical hy-

persurface is the product of an (n − r)-dimensional linear subspace E ^n−r

and a properly (r − 1)-dimensional quadric hypersurface. In case (II), the

quadric cylindrical hypersurface is the product of an (n − r − 1)-dimensional

linear subspace and a properly r-dimensional quadric hypersurface.

Let S ^p (r) denote the hypersphere in E ^p+1 with radius r and centered at the origin. Denote by M p,q the product of spheres

S ^p

r p

p + q



× S ^q

r q p + q



⊂ S ^p+q+1 (1) ⊂ E ^p+q+2 .

We denote by C p,q the (p + q + 1)-dimensional cone in E ^p+q+2 with vertex at the origin shaped on M p,q . It is easy to see that C p,0 and C 0,q are hyperplanes in E ^p+2 and E ^q+2 , respectively, and C p,q with p > 0, q > 0 are algebraic hypersurfaces of degree 2.

The purpose of this article is to prove the following classification theorem.

Theorem. A quadric hypersurface M in E ⁿ⁺¹ is of finite type (even locally) if and only if it is one of the following hypersurfaces:

(a) hypersphere,

(b) one of the algebraic cones C p,n−p−1 , 0 < p < n − 1,

(c) the product of a linear subspace E ^l and a hypersphere of E ^n−l+1 (0 < l < n),

(d) the product of a linear subspace E ^l and one of the algebraic cones C p,n−l−p−1 (0 < p < n − l − 1).

3. Properly n-dimensional quadric hypersurfaces. Let M be a hypersurface in E ⁿ⁺¹ . Consider a parametrization

(3.1) X(u 1 , . . . , u n ) = (u 1 , . . . , u n , v) where

(3.2) v = v(u 1 , . . . , u n ) . Denote ∂ i v(= ∂v/∂u i ) by v i . Then we have

(3.3) g ij = δ ij + v i v j , g ^ij = δ ij − v i v j

g where

(3.4) g = det(g ij ) = 1 +

n

X

i=1

v _i ² , and g ij = h∂ i X, ∂ j Xi. The Laplacian ∆ of M is given by

(3.5) ∆ = − X

i,j

 ∂ i g

2g g ^ij + ∂ i g ^ij



∂ j − X

i,j

g ^ij ∂ i ∂ j .

If M is a properly n-dimensional quadric hypersurface, then either M is

an algebraic cone of degree 2 or M is of one of the following two kinds:

v ² =

n

X

i=1

b i u ² _i + c , b 1 . . . b n c 6= 0 , (I)

v = 1 2

n

X

i=1

b i u ² _i , b 1 . . . b n 6= 0 . (II)

In the following two sections, we study properly n-dimensional quadric hypersurfaces of kinds (I) and (II), separately.

4. Proper quadric hypersurfaces of kind (I). In this section we assume M is a properly n-dimensional quadric hypersurface of kind (I). We may consider the following parametrization:

(4.1) X = (u 1 , . . . , u n , v) , v ² = a 1 u ² ₁ + . . . + a n u ² _n + c , a 1 . . . a n c 6= 0 . In this case, we have

(4.2) v i = ∂ i v = a i u i /v . Thus, (3.3) and (3.4) imply

(4.3) g ij = δ ij + a i a j u i u j

W , g ^ij = δ ij − a i a j u i u j

gW ,

(4.4) g = 1 + 1 W

X

i

(a i u i ) ² , 1

g = 1 − 1 gW

X

i

(a i u i ) ² , where

(4.5) W = v ² = a 1 u ² ₁ + . . . + a n u ² _n + c . From (4.4) we find

(4.6) ∂ i g = 2

W (a i u i (1 + a i − g)) ,

(4.7) e g := gW = c + X

i

(1 + a i )a i u ² _i . We put

A k = 1 2W

n

(gW − a ² _k u ² _k )∂ k g − a k u k

X

t6=k

a t u t ∂ t g o (4.8)

= 1 2 g X

t

g ^tk ∂ t g .

Then from (4.3) and a straightforward computation, we have

(4.9) − X

t

∂ t g ^tk = a k u k

gW X

t6=k

a t + 2A k

g ² .

From (3.5), (4.8) we obtain (4.10) ∆ = 1

g ² X

i

A i ∂ i + 1 gW

X

j

 X

t6=k

a t



a j u j ∂ j − X

i,j

g ^ij ∂ i ∂ j . We put

(4.11) c ij = gg ^ij .

From (4.3), (4.4) and (4.11) we have (4.12) c ij = δ ij + 1

W

 δ ij

X

t

a ² _t u ² _t − a _i a j u i u j

 . For later use, we note that from (4.8), (4.12) we have

(4.13) X

i,j

c ij (∂ i g)(∂ j g) = 2 X

j

A j ∂ j g . Also note from (4.7) that

(4.14) e g = gW is a polynomial in u 1 , . . . , u n . Lemma 1. We have

∆ ^t u k = g ^1−3t A k α t

 X

i

A i ∂ i g  t−1

+ g ^2−3t P k,t (u 1 , . . . , u n , 1/W ) where P k,t is a polynomial in n + 1 variables and α t is given by (4.15) α t = (4 − 3t)(6t − 5)α t−1 , α 1 = 1 .

P r o o f. The proof goes by induction. For t = 1, the formula follows from (4.10). Suppose the lemma is true for t − 1. Then it follows from (4.10), (4.11) and (4.13) that

∆ ^t u k = g ^1−3t X

j

A j A k α t−1

 X

i

A i ∂ i g  t−2

(4 − 3t)∂ j g

− g ^1−3t X

i,j

c ij A k α t−1

 X

l

A l ∂ l g

 t−2

(4 − 3t)(3 − 3t)∂ j g∂ i g + g ^2−3t P k,t (u 1 , . . . , u n , 1/W )

= g ^1−3t A k α t

 X

i

A i ∂ i g  t−1

+ g ^2−3t P k,t (u 1 , . . . , u n , 1/W ) , which proves the lemma.

Now, suppose that M is of k-type. Then there exist real numbers c 1 , . . . , c k such that

(4.16) ∆ ^k+1 X + c 1 ∆ ^k X + . . . + c k ∆X = 0 ,

(4.17) ∆ ^k+1 u i + c 1 ∆ ^k u i + . . . + c k ∆u i = 0 , i = 1, . . . , n .

From Lemma 1 and (4.17) we get

(4.18)  X

i

A i ∂ i g

 k+1

= gP (u 1 , . . . , u n , 1/W ) , where P is a polynomial in n + 1 variables. We put (4.19) G(u 1 , . . . , u n ) = W ⁵ X

i

A i ∂ i g .

Then G is a polynomial in u 1 , . . . , u n . Since W is a polynomial in u 1 , . . . , u n , there is a natural number N and a polynomial R in n variables such that (4.20) W ^N P (u 1 , . . . , u n , 1/W ) = R(u 1 , . . . , u n ) .

From (4.7), (4.18)–(4.20), we have

(4.21) W ^{N +1} G ^k+1 = e gW ^5k+5 R .

For any fixed j, 1 ≤ j ≤ n, we put u i = 0 for i 6= j in (4.21) to obtain (4.22) (c + a j u ² _j ) ^{N +k+2} 2 ^k+1 (a ² _j cu j ) ^2k+2

= (c + a j (a j + 1)u ² _j )(c + a j u ² _j ) ^5k+5 R(0, . . . , 0, u j , 0, . . . , 0) . Since a 1 . . . a n c 6= 0, this implies a j = −1. Because this is true for any j, M is a hypersphere.

5. Proper quadric hypersurfaces of kind (II). For such hypersur- faces we consider a parametrization

(5.1) X = (u 1 , . . . , u n , v) , v = 1 2

X

i

b i u ² _i , b 1 . . . b n 6= 0 . From (3.3)–(3.5) we may find

(5.2) g ij = δ ij + b i b j u i u j , g ^ij = δ ij − b i b j u i u j

g ,

(5.3) g = det(g ij ) = 1 + X

i

b ² _i u ² _i ,

(5.4) ∆ = 1 g ²

X

j

n

b j + X

i

(b j − b i )b ² _i u ² _i o

b j u j ∂ j

− X

i,j

g ^ij ∂ i ∂ j + 1 g

X

j

 X

i6=j

b i



b j u j ∂ j . Lemma 2. We have

(5.5) g ² ∆g = Q(u 1 , . . . , u n ) + gT (u 1 , . . . , u n ) ,

(5.6) ||∇g|| ² = 2

g Q(u 1 , . . . , u n ) ,

where Q and T are some polynomials in u 1 , . . . , u n and ∇g is the gradient of g.

P r o o f. From (5.3) and (5.4) we find

∆g = 2 g ²

X

j

b ² _j u j

n

(b j + X

i

(b j −b _i )b ² _i u ² _i )b j u j +g  X

i6=j

b i

 b j u j

o −2 X

j

b ² _j g ^jj . Thus, if we put

Q = 2 X

j

b ³ _j u ² _j n

b j + X

i

(b j − b _i )b ² _i u ² _i o , (5.7)

T = 2 X

j

b ³ _j u ² _j  X

i6=j

b i



− 2g X

i

g ⁱⁱ b ² _i , (5.8)

then we obtain (5.5). It is obvious that Q and T are polynomials in u 1 , . . . , u n . (5.6) follows from the definition of the norm of ∇g, (5.2), (5.3) and (5.7).

Lemma 3. We have

∆ ^t u j = g ^1−3t Q ^t−1 b j u j

n

b j + X

i

(b j − b _i )b ² _i u ² _i o

α t + g ^2−3t P e j,t

where e P j,t is a polynomial in u 1 , . . . , u n and α t is given by (4.15).

P r o o f. The proof goes by induction. For t = 1 the formula follows easily from (5.4). Assume it is true for t − 1. Then we have

∆ ^t u j = ∆ n

g ^4−3t Q ^t−2 b j u j



b j + X

i

(b j − b _i )b ² _i u ² _i 

α t−1 + g ^5−3t P e j,t−1

o

= g ^1−3t Q ^t−2 b j u j



b j + X

i

(b j − b i )b ² _i u ² _i  α t−1

× {(4 − 3t)g ² ∆g − (4 − 3t)(3 − 3t)g||∇g|| ² } + g ^2−3t P b j,t .

where b P j,t is a polynomial in u 1 , . . . , u n . Thus, Lemma 2 implies the asser- tion.

If M is of k-type, then again there exist real numbers c 1 , . . . , c k such that

∆ ^k+1 u j + c 1 ∆ ^k u j + . . . + c k ∆u j = 0 , j = 1, . . . , n . From Lemma 3 and (5.7) we obtain

Q ^k+1 = gP (u 1 , . . . , u n )

where P is a polynomial in u 1 , . . . , u n . Since b 1 . . . b n 6= 0, g = 1 + P b ² _i u ² _i

is irreducible. Moreover, because Q/g = ¹ ₂ ||∇g|| ² is not a polynomial in

u 1 , . . . , u n , we obtain a contradiction. Thus, there exist no proper quadric

hypersurfaces of kind (II) which are of finite type.

6. Proof of Theorem. If M is a properly n-dimensional quadric hy- persurface of finite type in E ⁿ⁺¹ , then either M is an algebraic conic hyper- surface of degree 2 or, according to §§3–5, M is a hypersphere. If M is an algebraic conic hypersurface of degree 2, then because M is of finite type, M is a minimal cone [G]. Thus, by a result of [H], M is one of the algebraic cones C p,n−p−1 , 0 < p < n − 1.

If M is a quadric cylindrical hypersurface of finite type in E ⁿ⁺¹ , then M is the product of a linear subspace E ^l and a proper quadric hypersurface, say N . Since M is of finite type, N is also of finite type. Thus, N is either a hypersphere or an algebraic cone C p,n−l−p−1 for some suitable p.

The converse is easy to verify.

REFERENCES

[C1] B. Y. C h e n, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore 1984.

[C2] —, Surfaces of finite type in Euclidean 3-space, Bull. Soc. Math. Belg. S´ er. B 39 (1987), 243–254.

[CD] B. Y. C h e n and F. D i l l e n, Quadrics of finite type, J. Geom. 38 (1990), 16–22.

[CDVV] B. Y. C h e n, F. D i l l e n, L. V e r s t r a e l e n and L. V r a n c k e n, Ruled surfaces of finite type, Bull. Austral. Math. Soc. 42 (1990), 447–453.

[D] F. D i l l e n, Ruled submanifolds of finite type, Proc. Amer. Math. Soc. 114 (1992), 795–798.

[G] O. J. G a r a y, Finite type cones shaped on spherical submanifolds, ibid. 104 (1988), 868–870.

[H] W. Y. H s i a n g, Remarks on closed minimal submanifolds in the standard Rie- mannian m-sphere, J. Differential Geom. 1 (1967), 257–267.

DEPARTMENT OF MATHEMATICS DEPARTEMENT WISKUNDE

MICHIGAN STATE UNIVERSITY KATHOLIEKE UNIVERSITEIT LEUVEN EAST LANSING, MICHIGAN 48824-1027 CELESTIJNENLAAN 200B

U.S.A. B-3001 LEUVEN, BELGIUM

DEPARTMENT OF MATHEMATICS HENAN UNIVERSITY

KAIFENG, HENAN 475001 PEOPLE’S REPUBLIC OF CHINA

Re¸ cu par la R´ edaction le 30.8.1990