A THEOREM OF PICK-BERWALD TYPE FOR A TOTALLY UMBILICAL AFFINE IMMERSION OF GENERAL CODIMENSION

A THEOREM OF PICK{ BERWALD TYPE

FOR A TOTALLY UMBILICAL AFFINE

IMMERSION OF GENERAL CODIMENSION

Naoyuki Koike and Yoko Torisawa

(Received October 26, 1998)

Abstract. It is known that a totally umbilical ane immersion of general

codimension into an ane space is anely congruent to a graph immersion or a centro-ane immersion. In this paper, we shall investigate a more detailed property of such an immersion.

AMS1991 Mathematics Subject Classi
cation. 53A15.

Key words and phrases. totally umbilical ane immersion, ane shape tensor, ane fundamental form, hyperquadric.

Introduction

Throughout this paper, unless otherwise mentioned, we assume that all objects are of classC

1 and all manifolds are connected ones without

bound-ary. Also, denote by ;(E) the space of all cross sections of a vector bundle E. An ane (or equiane) immersion of codimension one into the (n+

1)-dimensional ane space (

R

n+1

r
~ !~) with the natural equiane structure has

been studied by some geometricians. In particular, if an equiane immersion into (

R

n+1

r
~ !~) satises the volume condition, then it is called a Blaschke

immersion (see 1], 5]). For a Blaschke immersion, G. Pick and L. Berwald proved the following result (see 5]):

If f is a Blaschke immersion of an n(
2)-dimensional manifold (M
r
)

with equiane structure into (

R

n+1

r
~ !~) and its cubic form (i.e., the

co-variant dierentiation of its ane fundamental form) vanishes, then f(M) is

contained in a hyperquadric in

R

n+1.

Here we note that such an immersion is totally umbilic (see 5]). On the other hand, in 3], K. Nomizu and U. Pinkall proved the following characterization theorem for a totally umbilical ane immersion of general codimension into an ane space with the natural torsion-free ane connection:

Letf be a totally umbilical ane immersion of ann-dimensional manifold

(M
r) with torsion-free ane connection into the (n+r)-dimensional ane

space (

R

n+r

r~) with the natural torsion-free ane connection, where n
2

andr
1. Thenf is anely congruent to a graph immersion or a centro-ane

immersion.

Here a graph immersion is dened as follows. LetF be an

R

r-valued function

on then-dimensional ane space

R

nand f an immersion of

R

ninto (

R

n+r
r~) dened by f(x) = (x
F(x)) 2

R

n 

R

r =

R

n+r ( x 2

R

). Let N be the

transversal bundle along f such that N x (

x 2

R

n) are parallel to the ane

subspace

R

rof

R

n+r. Denote by

rthe induced connection on

R

nfor N. Then f is an ane immersion of (

R

n
r) into (

R

n+r

r~). Such an ane immersion is

called a graph immersion. Note that its ane shape tensor vanishes identically. Also, a centro-ane immersion is dened as follows. Letf be an immersion of

ann-dimensional manifoldM into (

R

n+r
r~) admitting an (r;1)-dimensional vector subspace Vof

R

n+r such that f ( T x M)Spanff(x)gV =

R

n+r

holds for everyx2M, whereT x

M is the tangent space ofM at x, we identify T

f(x)

R

n+r with

R

n+r and

f(x) is its position vector. Dene a tansversal

bundle N along f by N

x = Span

ff(x)gV (x 2 M). Denote by r the

induced connection onM forN. Thenf is an ane immersion of (M
r) into

(

R

n+r

r~). Such an ane immersion is called the centro-ane immersion.

Note that its ane shape tensor A does not vanish and that V = Ker x

(x2M) holds, where  is a cross section ofN  with

A=I.

In this paper, we shall prove the following result similar to the Pick-Berald theorem for a totally umbilical ane immersion of general codimension.

Theorem.

Letf be a totally umbilical ane immersion of an n-dimensional

manifold (M
r) with torsion-free ane connection into the (n+r)-dimensional

ane space (

R

n+r

r~) with the natural torsion-free ane connection, where n
2 andr
1. If its ane shape tensorAdoes not vanish and the covariant

dieretiationrof its ane fundamental form vanishes identically, thenf(M)

is contained in a cylinder over a hyperquadric in an (n+1)-dimensional ane

subspace of

R

n+r.

Here we note thatf(M) is not necessarily contained in an (n+1)-dimensional

ane subspace of

R

n+r in spite of being totally umbilic and

r= 0. In fact,

according to the reduction theorem for an ane immersion of K. Nomizu and U. Pinkall (see 3]), the conditionr= 0 implies that the dimension of its rst

normal space N 1

x at

x(i.e., the linear span of the image of

x) is independent

of the choice of x 2M and that f(M) is contained in an (n+s)-dimension

ane subspace of

R

n+r, where

s = dimN 1

x, but the totally umbilicity of f

does not necessarily imply dim N 1

x = 1. In

x3, we shall give an example of a

totally umbilical ane immersion as in the statement of Theorem such that the dimension of its rst normal space is more than one.

x

1. Fundamental formulas and denitions

In this section, we shall recall the fundamental formulas and denitions for an ane immersion. Let (M
r) (resp. ( ~M
r~)) be an n (resp. (n+ r))-dimensional manifold with torsion-free ane connection. An immersion f : (M
r) ,! ( ~M
r~) is called an ane immersion if there is a transversal

bundle N along f such that for every tangent vector elds X and Y on M,

~ r X f  Y ;f ( r X

Y) is a cross section of N. Note that the choice of such a

transversal bundle N in general is not unique. In the sequel, we x such a

bundleN. Set (X
Y) := ~r X f  Y ;f ( r X Y):

This quantity  becomes an N-valued symmetric tensor eld of type (0
2)

on M. This tensor eld  is called the ane fundamental form of f. For a

transversal vector eld  along f (i.e., 2;(N)), we write

~ r X =;f ( A
X) +r ? X 
where A
X 2 ;(TM) and r ? X

 2 ;(N). This quantities A becomes a cross

section of the tensor product bundle N  T  M TM and r ? becomes a connection onN, whereN (resp. T 

M) is the dual bundle ofN (resp. TM).

This tensor eldA is called the ane shape tensor of f and r

? is called the

transversal connection of f. The covariant dierentiationr of  is dened

by (r X )(Y
Z) :=r ? X( (Y
Z));(r X Y
Z);(Y
r X Z)

for X
Y
Z 2 ;(TM). The ane immersion f is said to be totally umbilic if

there is 2;(N

) with

A=I, whereI is the identity transformation of TM.

x

2. Proof of Theorem

In this section, we shall prove Theorem stated in Introduction. Proof of Theorem. Let  be a cross section ofN

 with

A=I. Sincef is

totally umbilic andA6= 0,f is anely congruent to a centro-ane immersion

from the Nomizu-Pinkall theorem. Hence, the transversal space N x of f at x2M is decomposed as follows: N x = Span ff(x)gKer x :

Now we dene a hypersurfaceF x ( x2M) in

R

n+r as F x:= ff(x) +f ( U) + f(x) +jU 2T x M
2

R

2Ker x
 x( U
U);( 2+ 2 )f(x) 0 (mod Ker x) g:

Fig. 2.

We show that F

x is a cylinder over a hyperquadric in an (

n+ 1)-dimensional

ane subspace of

R

n+r. Let p 1 (resp. p 2) be the projection of N x onto Ker  x

(resp. Spanff(x)g). Then it is easy to show that x(

U
U);( 2+2

)f(x) 0

(mod Ker

x) holds if and only if

(2.1) p 2(  x( U
U)) = ( 2 + 2 )f(x)

holds. We dene a symmetric bilinear form h onT x M by p 2(  x( X
Y)) =h(X
Y)f(x) forX
Y 2T x M. Then (2:1) is equivalent to (2.2) h(U
U) = 2 + 2 :

Let (e 1

e n) be a basis of T x M. We putU = n P i=1 U i e i and h ij = h(e i
e j) (i
j = 1

n). Then (2:2) is rewritten as (2.3) n X ij=1 h ij U i U j = 2 + 2 : Let= (y 1

y

n+1) be the ane coordinate system of the (

n

+1)-dimension-al ane subspacef (

T

x

M)Spanff(x)gassociated with the basis (f  e 1

f  e n

f(x)), where the origin is the pointf(x). Setv:=f(x)+f ( U)+ f(x). Then we have (v) =(f(x) + n X i=1 U i f  e i+ f(x)) = (U 1

U n
)
that is, y i( v) =U i ( i= 1

n)
y n+1( v) = : Hence (2:3) is rewritten as n X ij=1 h ij y i( v)y j( v) =y n+1( v) 2 + 2y n+1( v):

Therefore, by noticing that Ker is parallel with respect to ~r, we see thatF x

is a cylinder over a hyperquadric n P ij=1 h ij y i y j = y 2 n+1+ 2 y n+1 in the ( n+

1)-dimensional ane subspacef (

T

x

M)Spanff(x)g of

R

n+r.

Now we shall show thatf(M) is contained inF x. Fix x 0 2M andz 0 2F x 0.

We dene a tangent vector eld ~U onM, a function ~ onM and a transversal

vector eld ~ on M satisfying ~ x 2Ker x ( x2M) by z 0 = f(x) +f (~ U x) + ~ x f(x) + ~ x ( x2M):

Fig. 3.

Then we have ~ r X z 0 = (~ + 1)f ( X) +f ( r X~ U) +(X
U~) + (X ~)f+r ? X~ = 0

for every tangent vector eldX on M. By taking notice of the tangent

com-ponent and the transversal comcom-ponent of this equation, we have

(2.4) r X~ U =;(~ + 1)X and (2.5) (X
U~) =;(X ~)f ;r ? X~ :

Now we dene the transversal vector eld  onM by

 =( ~U
U~);(~ 2+ 2~

)f:

From r= 0, (2:4) and (2:5), we have r ? X = r ? X( ( ~U
U~));r ? X((~ 2+ 2~ )f) = 2(r X~ U
U~);2(~ + 1)(X ~)f ;(~ 2+ 2~ )r ? X f =;2(~ + 1)(X
U~);2(~ + 1)(X ~)f = 2(~ + 1)(X ~)f+ 2(~ + 1)r ? X~ ;2(~ + 1)(X ~)f = 2(~ + 1)r ? X~ :

Since Ker is parallel with respect to r

?, we have r ? X~ 2 Ker. Hence we have r ? X 0 (mod Ker): It follows fromz 0 2F x

0 and the denitions of ~

U and ~ that x 0 = (~U x 0
U~ x 0) ;(~ 2 x0+ 2~ x 0) f(x 0) 0 (mod Ker x 0) : Hence we have x= (~U x
U~ x) ;(~ 2 x+ 2~ x) f(x) 0 (mod Ker x)

for everyx2M. Therefore, we can obtain z 0 = f(x) +f (~ U x) + ~ x f(x) + ~ x 2F x ( x2M):

This together with the arbitrariness of z

0 deduces F x 0  F x ( x 2 M).

Fur-thermore, from the arbitrariness ofx 0 and

x, we haveF x0 =

F

f(x) 2 F

x0 because of

f(x) 2 F

x. After all, from the arbitrariness of x, we

can obtainf(M)F x

0. This has completed the proof.

x

3. An example

In this section, we shall give an example of a totally umbilical ane im-mersion as in the statement of Theorem such that the dimension of its rst normal space is more than one.

Example. Dene a map f from

R

n to

R

n+r by f(x 1

x n) := ( x 1

x n
a 1 x 2 1+ b 1 x 1+ c 1

a r 0x 2 r 0+ b r 0x r 0+c r 0
0

0
c) ((x 1

x n) 2

R

n)
wheren
2
r
3
r 0:= min fn
r;1g
a i( i= 1

r 0) and care non-zero constants, and b i and c i ( i = 1

r

0) are constants. Also, dene a map N

from

R

n to the Grassmann manifold G rn+r of all r-dimensional subspaces of

R

n+r by N (x1xn) := Span f @ @y n+1

@ @y n+r;1
f(x 1

x n) g ((x 1

x n) 2

R

n)
where (y 1

y

n+r) is the natural coordinate system of

R

n+r and f(x

1

x

n) is its position vector. Easily we have f ( @ @x i ) = @ @y i + (2a i x i+ b i) @ @y n+i (i= 1

r 0)
f ( @ @x j ) = @ @y j (j=r 0+ 1

n)
f(x 1

x n) = n X j=1 x j @ @y j + r 0 X j=1 (a j x 2 j + b j x j+ c j) @ @y n+j +c @ @y n+r :

From these relations, we can show the linearly independence off ( @ @x 1)

f ( @ @xn)
@ @yn+1

@ @yn+r;1
f(x 1

x n). That is, f is an immersion andN

is regarded as a transversal bundle alongf. Let ~rbe the natural torsion-free

ane connection of

R

n+r and

rthe induced connection on

R

n for

SinceN = Spanf @ @yn+1

@ @yn+r;1 gSpanff(x 1

x n) gand Spanf @ @yn+1

@ @y n+r;1

gis parallel with respect to ~r, the immersionf is a centro-ane

immersion of (

R

n

r) into (

R

n+r

r~). That is, the immersion f is totally

umbilic and its ane shape tensor does not vanish. Concretely its ane shape tensorA is given byA=I, where is the cross section of N

 dened by (f(x 1

x n)) = ;1 and ( @ @y n+i) = 0 ( i= 1

r;1). Also, we have ~ r @ @x i f ( @ @x i ) = 2a i @ @y n+i (i= 1

r 0)
~ r @ @x i f ( @ @x i ) = 0 (i=r 0+ 1

n)
r~ @ @x i f ( @ @x j ) = 0 (1i6=jn) and hence ( @ @x i
@ @x i ) =2a i @ @y n+i (i= 1

r 0)
( @ @x i
@ @x i ) = 0 (i=r 0+ 1

n)
( @ @x i
@ @x j ) = 0 (1i6=j n)
r @ @x i @ @x j = 0 (i
j= 1

n):

Thus its rst normal space is spanned by @ @yn+1

@ @y n+r 0 at each point of M, that is, its dimension is equal tor

0 (
2). Also, we have (r @ @x i )( @ @x j
@ @x k ) =r ? @ @x i (( @ @x j
@ @x k ));(r @ @x i @ @x j
@ @x k ) ;( @ @x j
r @ @x i @ @x k ) = 0 (i
j
k = 1

n)

which impliesr = 0. Thus this ane immersionf : (

R

n

r),! (

R

n+r

r~)

is a desired totally umbilical ane immersion.

Acknowledgement

The authors would like to express their sincere gratitude to Professor S. Yamaguchi for his helpful advice and to Professor N. Abe for his constant encouragement.

References

1] Blaschke, W., Vorlesungen uber Dierentialgeometrie II, Ane Dierentialgeometrie,

Springer, Berlin, 1923.

2] Erbacher, J., Reduction of the codimension of an isometric immersion, J. Dierential Geometry5(1971), 333-340.

3] Nomizu, K. and Pinkall, U., Cubic form theorem for ane immersions, Results in Math.

13(1988), 338-362.

4] Nomizu, K. and Sasaki, T., Centroane immersions of codimension two and projective hypersurface theory, Nagoya Math. J.132(1993), 63-90.

5] Nomizu, K. and Sasaki, T., Ane dierential geometry, Cambridge University Press, 1994.

6] Simon, U., Schmenk-Schellschmidt, A. and Viesel, H., Introduction to the ane dier-ential geometry of hypersurfaces, Lecture Notes, Science university of Tokyo, 1991. Naoyuki Koike

Department of Mathematics Faculty of Science Science University of Tokyo Wakamiya-cho 26, Shinjuku-ku Tokyo 162-0827, Japan

Yoko Torisawa

Department of Mathematics Faculty of Science Science University of Tokyo Wakamiya-cho 26, Shinjuku-ku Tokyo 162-0827, Japan