Applications of Polarity (*) (**).

H. GUGGENn-ErSm~ (Brooklyn, U.S.A.)

To B~IA~I~o SEGR]~ on his 70-th birthday

Sunto. - Nella prima parte di questo lavoro si svituppa la geometria dif]erenziale del triedro mobile per il gruppo SL(n). La polarit5 nella s]era uuit5 t~'as]orma una matrice di 2'tenet C in -- C r per varie geometric liueari e proiettive. Nella seconda parte si deriva una serie di di- suguaglianze integrali basate sulla tras]ormazione per polarit& Nella terza parte si calcola la tras]ormazione della matrice Hessiana e della curvatura di Gauss in polarit~ e ne ]a ap- plicazione ai teoremi di rigidith di Liebmann-Hilbert e JSrgens-Calabi-t)ogorelov.

I. - D i f f e r e n t i a l g e o m e t r y o f l i n e a r g r o u p s a n d po lar i t i e s .

1. - The contac t t r ans format ion induced b y polar i ty in the elliptic paraboloid

known as Legendre ' s t r ans fo rmat ion has been used in the theory of differential equa- t ions since it was in t roduced b y EULER. The contact t r ans format ion induced b y

po la r i ty in the uni t sphere is basic to the theory of convex bodies and useful in problems of differential equations. I n this sense, we s tudy in this pape r several aspects of the analy t ic theory of polari ty, ma in ly in the uni t sphere. The differential

geomet ry developed here has some points of contact wi th the theories developed b y P. and A. SCHI~O~OW [11] chap. IV. , bu t wi th a different emphasis.

I n this section, we develop the differential geomet ry of moving f rames of hyper-

surfaces in E ' - - 0 for the act ion of the group S L ( n ) = SL(n , R) of volume pre- serving l inear t ransformat ions of E n. For the local theory, we give a surface by a

m a p X : V - - > E ~ - - O of some coordinate neighborhood V c E ~-1. The vector of the poin t X is again denoted b y X . W e consider only local sur]aces /or which no tangent

p lane passes through the origin.

An adapted ]tame of X for SZ(n ) is a f rame (X , el~ ...7 e,_~) for which

det (X , el, ..., e,_l) -= !

and the es are t angen t to the surface a t X. The Frene t equations of this f rame will

(*) Entrata in Redazione il 18 settembre 1974. (**) Research partially supported by NSF Grant GP-27960. AMS (MOS) classification:

26 A 84, 28 A 86, 35 A_ 30, 35 K 99, 53 A 15, 53 A 20.

370 H. GUGGE~HE~fE~: Applications o/ polarity

be of the form

a

(X

el

len_l

0 0 . 1 . . . (~--II X

el

I n the following, indices will a lways run f rom 1 to n - - 1 . I t is convenient to

introduce the row vectors of l inear differential forms a = (a~), a * = (a*) and the

also denote l inear (exterior) differential column vector of vectors e = (e~). The co~ forms.

The condition on the de te rminan t becomes

~ = 0 (2) .

The Poinear6 relat ion for co ° = 0 gives

= 0

and this means, by Car tan ' s theorem, t h a t there exists a symmet r ic ma t r ix W---- W T for which

a = a W .

The unimodular t ransformat ions of the f rames t h a t t r ans form adap ted f rames into adap ted f rames are of the fo rm

r 1

0

• • • 0 '

A A ~ S L ( n - - 1 ) .

For the vectors e', a ' , a* ' and the ma t r ix W' of the t r ans fo rmed sys tem we have a '= aA, e - Ae' and, by differentiation,

a * ' - - a* (A-~)T = o" W'

H. GUGGE~KEIY£ER: Applications o] polarity 371

i . e . ,

(3) W = A W ' A T

the matrix W represents a quadratic/orm which we m a y call the SL-]undamental ]orm of X : V - > E " - 0. The invar ian ts of W (determinant , rank, signature) for unimo-

dular t rans format ions ~re un imodular invar ian ts of X. A point of X is classified as elliptic, parabolic, or hyperbol ic corresponding to W being the ma t r i x of a definite, degenerate , or indefinite nondegenera te form. The invar ian t

A = det W

is of major impor tance . The vector p roduc t c = [a~, ..., a ._l] of n - - 1 vectors al , ..., a._~ in E" is the

unique vector for which

det (al, ..., a ._ l , x) = (c, x)

for a rb i t r a ry x. Here (u, v) is the scalar p roduc t of the euclidean s t ruc ture induced

b y the car tes ian s t ruc ture of E ~. The m a p X : V-+~" - -O induces X*: V-+E~--O b y

(4) X* = [e~, ..., e._~].

Since any two adap ted f rames are t r ans formed into one another by a t r ans format ion of SL(n- -1 ) , the definition of X* does not depend on the par t icular adap ted f rame e.

Since (X, X * ) = 1, the surface X* is the polar reciprocal of X in the uni t sphere. I f X is ~ smooth , convex surface wi th an a d a p t e d f rame, the po la r i ty of convex bodies induces a o n e - - t o - - o n e m a p of the bounda ry surface onto the bounda ry of

the polar surface. I n t h a t case, the induced m a p X - > X * is given b y (4). I f A ~e 0, an adap ted f rame of X* is obta ined as

c~ = - - [el, ..., ei_l, X, e~+l ~ ... ~ e~_l] •

since, b y (1),

dX* = ~ [ e l , ..., el-l, dc,, e,+l, ..., e~_l] i

= •

i

The f rames (X, e) and (X*, e*) are reciprocal since

(X*, X) = 1

(e*, e~) = - - d e t (ei, el, ..., ei_l, X, ei+l, ..., e~_~)

~iJ ,

( x * , e,) = ( x , = o .

372 H. GUGGENttEI~iER: Applications o] polarity

Hence, also

( dX, * ) + (X, de*) = 0

(de*, ej) + (e*, de~) = 0

and the F renen t formulas for the polar reciprocal surface are

(5)

)t ° w O - i

$

- 4 .

The in tegrabi l i ty conditions of (1) and (5) are identical:

(6) d(~* ~ *

We formula te ore' results on the act ion of polar i ty in S~-1:

PtcOPOSI~IO~ 1. - I] a sat]ace X in E " - - 0 admits an adapted/tame ]or the ac- tion o] SL(n) then the polarity in the unit sphere induces a smooth point map X -+ X*. For the matrix C appearing in the Frenet ]ormula, the action o] the polarity is given by

C* = C(X*) = - - C(X) r .

A par t ia l result for surfaces in E 3 was previously obta ined by E. SALKOWSXI [10]. The map X -> X* is induced b y a l inear map if C, + C T-- 0. I n t ha t case, X is

the l inear image of a sphere, i.e., an ellipsoid.

I t follows f rom (!) t h a t alA.. .Aa ~-I is n t imes the volume e lement of a cone of center 0 and basis the paral le lotope of edges ale1, ...,a"-le._~. For A=/=0, the

n - - 1 * * volume element spanned b y an area e lement of X* is ( --1) a 1A...Aa~_ 1 t imes 1/n. Hence, A(:/: 01 is the rat io of the two volume elements t h a t correspond to one another in the polar i ty . I n t e rms of the euclidean data ,

(7) A = Kh -"-~

where K is the Gauss curva tu re and h the suppor t dis tance (of the t angen t plane f rom the origin) [6].

H. GUGGEI~n~VM~,R: Applications o] polarity 373

4 . - As an application of the theory of the preceding section, we discuss some aspects of the unimodular linear geometry of an elliptic surface in E 3. We assume that the surface is concave towards the origin. Since W is in this ease congruent to the identity matrix, there exist frames fox" which (1) has the particular form

(8) tee:~ t - - 0 a ~

\ - - A~ a ~ ~

¢O e I .

These frames are called good ]tames. Data given for good frames will be indicated by bold face letters. Any two good frames are transformed into one another by a rotation

cos~ s ina i A = \ _ s i n 9 cosq]"

In such a rotation, the elements of the matrix C are changed by

a 1 --> a 1 cosq~ + a 2sin

a ~ -->-- a I sin ~v + a ~ cos 9

to~-+ ¢o]cos2q+ ½(to~-4- ~ ) s i n 2 9

~ ~ - - ¢o~ sin 2 ~ - - to~ s in 2 ~v + to~ c o s ~ + d ~

- s m + c o s % - s i n -

and hence, the function U(X) defined by

togA (¢o~ 4- to~) = U(X)(~I Aa 2

is an invariant of the geometry. We say that a point X is an SL-umbilic if U(X) = O.

PROPOSITIO~ 2. -- I] all points o] a neighborhood are SL-umbilics and i] A is con- stant in that neighborhood then the neighborhood is on an ellipsoid of center 0 i] it is concave towards the origin, on a two-sheeted hyperboloid i] it is convex towards the origin. The induced Riemannian metric ds ~ = (al)~+ (a2) 2 is o] constant positive curvature in the ]irst case, o/ eonstant negative curvature in the second case.

Since A is supposed to be constant, we may assume A = 1 after a homothety. $

Equations (6) for a~ = ± a TM give

374 H. GU(~GENI-mI~m Applicat ions of polarity

1 = ia~ @ j~e, these equations give ¢o~ ~- ¢o~ = - - j ~ l -t- I a e. Then If we write ~ U = 1 ~ - J~ and by our hypothesis ¢o~= ¢o~-t- ¢o~ = 0. The ma t r ix of (8) becomes

e~ ~ 0 .

\ ~ t $ 2 ~ tO 1

I f the surface is concave towards the origin, e = - - 1 and the mat r ix is skew: i t is the infinitesimal mat r ix of an orthogonM mat r ix funct ion t imes a constant non- singular matr ix . Since X is the first vector of t ha t mat r ix funct ion i t is the affine image of the uni t sphere of center O. I f the surface is convex towards the origin, e = - t - 1 and the mat r ix is in the Lie algebra of the automorphism group of the hyperboloid of two sheets (cf. [3], p. 301). In any case, the s ta tement about the

Gauss curvature follows from dco~ = e~lAa ~ (cf. [3], (10)-(20)). In the same way, one may develop the general theory of surfaces. We only note

here tha t

(9) A* ---- A - t U* = UA -~ .

2. - I t is not difficult now to obtain the invariants of an hypersurface for the geometry of GZ(n, R) in E" ~ 0. ]?or this~ we have to s tudy the action of an homo- t h e t y on the frames adapted to SL(n , R) . For good frames on elliptic surfaces, we

have * ~ cAll(n-I)6 ~ a~ ~ e = : t : 1 .

The action of an homothe ty of rat io y t ransforms the frame (X~ e) into (yX~ ~-~/(~-~)e). Henc% for a eovariant f rame E =-(Eo, E~, ..., E~_~) we ma y choose

E o = Av2~X

The da ta in d E = CE, i.e., in

are connected with those of (1) by

~ = All~-.(n -1)a~

as-* ~ All(n-1)a~

-~ ~ ( i ~ j ) (10) ~o~ = o~

e)o0 = (2n)-I d logA

~ = ¢% ~ 2 n ( n - - 1)-1d logA

H. GUGGENttEIlVLER: Applications o] polarity 375

which again yields

(11)

for the polar frame

C* ----- -- C r

~ ' = [E, , ..., E,_, , ~o, E,÷, , ..., ~ ,_~] .

For a projective interpretation one has to consider a sufficiently small piece of surface as the projective lift of an aifine surface in E ~-~ (and several types of vector products adapted to the signature of the quadrie that defines the polarity.) For the study of polar surfaces, this approach to projective surface theory is more convenient than either that of Fubini-~ech or that of Bol. On the other h~nd, polarity is easily treated in the GL(n, 1¢) geometry developed by D. LAvGwI~z ([8], p. 39-42).

II . - A ser ies o f i n t e g r a l i n e q u a l i t i e s .

1. - In [4], the following problem (needed for the study of x"-~ px : 0) was solved: Given a locally convex arc with prescribed endpoints whose polar angle is monotone increasing from ~0 to ~1 and whose polar radius convers an area A (counting multiplicities). What is the curve for which the corresponding area A* of the polar in the unit circle is minimal? The minimal curve is the polygonal are satisfying the conditions and having a minimal number of vertices. The proof uses only that polar- ities in the plane map pencils of points onto pencils of lines and that the particular polarity in the unit circle maps convex arcs that arc concave towards the origin onto arcs of the same kind. The result of [4] therefore holds for all polarities in ellipses whose center is the origin.

A similar minimum problem was treated in [5]: The compact convex body of volume V which contains the origin and whose polar in the unit sphere has minimal volume V* is a polytope with a minimal number of vertices if the class of admissible bodies is restricted to a closed subspace of the closure (in the sense of the Blaschke topology) of the set of polytopes for which there exists an (n--1)-face whose normal makes angles <~/2 with the normals of the adjacent faces, and their potars. The angle conditions is imposed by the method of proof, not the problem.

We shall use the minimum properties established in [4, 5] to derive new integral inequalities. Compared with arguments about inscribed and circumscribed set (el. [1]), the construction of a minimal polytope improves the constants by factors between n and hi.

2. - THEOI~EI~. -- Let z ~-- ](xl, ..., xn) ---- ](x) be a C2-]unction on a convex domain G o] E" which contains the origin. The volume o] G is V(G) and that o] the polar G* o] G

376 H. GV~GENm~rE~: Applioations o] polarity

in the unit sphere is V(G*). I]

](x) > o

l(x) = o

]or x ~ int G

]or x e ~G

H(]) = (~]/(~xi~x~)) is a negative de]inite matrix ]or x ~ i n t G then

( - 1)of axe.., dxofg- d(p )... a(p o ) > (n + V(O) V(O*) G

where p~ = az/~x~ and g-= z - - ~ x ~ p ~ . I] G is symmetric about the origin, the product of the integrals is greater than 4~(n-~-1)! . The inequalities are sharp.

PROOF. - The surface z---- ](x) is concave in E ~+~. The surface together wi th its

reflection in the hyperp lane z - - 0 bounds a convex body K whose volume in E ~+~ is

V,+~(K) = 2 fzdxl ... dx~ . G

The vector of a poin t on the surface

X = (x~, . . . , x , , , z)

and the t angen t vectors

e1 = (1~ 0~ ..., 0~ Pl) , . , , , . , o o , ° . . . .

e, ---- (0~ 0~ ... ~ 1~ p . )

fo rm a basis of vectors in E "+~ for which

det (X, e~, ..., e,)----- ( - - 1 ) ~ g ¢ 0

since g is posi t ive on ~G and has a single critical point in G a t x = 0 where grad g = xTH vanishes since H is definite. I n addit ion, g(O) = z(O) > O. An adap ted

f l a m e for the surface is g iven b y X and the vectors e~g-~/ '~e~, i = 1 , ...~ n. The vector of the polar surface corresponding to X is

(1) X * = [c i , . . . , c~] = g - l ( _ p l , . . . , _ p , , 1 ) .

The bounda ry ~K* of the polar body K* consists of the two local surfaces given b y X* for z > 0 and z < 0 and a cylindrical surface over the basis ~G* wi th generators paral lel the z-axis. The vo lume of K * is

V~+I(K*) = 2(- - 1)~fg -1 d(plg -1) ... d(p,g-~) . G

H. GVaGE~H~rEa: Applications o] polarity 377

I n order to obta in the es t imates of the theorem, we have to find a lower bound for

the produc t V.+~(K)V.+~(K*) if the convex body K contains 0 and is symmet r i c for a hyperp lane th rough 0. B y the approx imat ion theorems for convex bodies~

we have to do this only for po ly topes G. B y the resul t of [8], the m i n i m u m is obta ined for a double p y r a m i d of basis G. The polar in the uni t sphere of E ~+~ of such a double

p y r a m i d of a l t i tude h is the r ight p r i sm of a l t i tude h -~ over G* (polar of G in :E*).

Hence,

V~+~(K) F~+~(K*)>~ 2 . hF(G).2h_~V(G. ) n--~±

4 - n + 1 ~ ' ( a ) V ( G * ) .

I f G is poin t symmet r i c of center 0, i t was shown in [8] t ha t V(G)V(G*)>4~/n!: The inequali t ies are s t r ic t since K is not a polytope. They are sharp for convex bodies

since t hey are sharp for polytopes .

3. - The inequal i ty of the theorem is re la t ively simple for n = 1, 2. In t ha t case it was shown [6] t h a t V(G)V(G*) has the sharp lower bounds 4 for n = 1

and 27/4 for n = 2 (in the case of u n s y m m e t r i c bodies.)

P ~ o P o s I ~ o ~ 1. - I] y(x) is a positive, concave C~-]unction on an inte~'val xo < x <~x which vanishes at the endpoints then~ /or any ~ (Xo, x,)

f f - Y ~ [ (x - ~ ) y - - y ] " 8 >/2.

The proposi t ion is an immedia te consequence of the theorem. The p roduc t of the integrals is invar ian t in a di lat ion y - ~ cy. An immedia te consequence is the following.

PlcoPosImxo~ 2. - .Let e(x) be the conjugate point o] x /or a di/]erential equation y" ~ p(x)y = O, p(x) > O. For any solution y o] the equation that vanishes at xo we have

C(Zo) c(~o) y(x) >

w , ~¢0

/or all ~ ~ (Xo, e(xo)).

:For n = 2, we obta in :

Ih~oPosITIO~ 3. - Zet z(x, y) be a C~-]unetion de/ined on a convex domain G and vanishing on ~G. I] the conditions

z > O , z~z w - z~,~,> 0 , z ~ < O

378 H. GUGGE~n'~.r~.V~: Applications o] polarity

ho~d in int G then, ]or all (xo, Yo)~int G we have

z dx dy z [ z - - (x - - xo) z~-- (y - - Yo) z~] ~ ~f"

The product o] the integrals is ~> 8/3 i / G is symmetric o] center (xo, Yo).

For n > 2, the explicit form of the second integral in the formula of the theorem is complicated. Let Sk be the k-th elementary symmetric function of the variables x~p~. We also put

c ~ = { j~ - 1 for k = 2 j

- - j for k = 2 j ~ - l .

A laborious induction the shows

~--1 ? x

( - 1)-g-id(plg-~)... d(pog-1) = g-(-+~, detH(~){~ + Z c~a-~S~,,) dx, ... d ,c . \

2 ]

We can obtain a more convenient expression by the methods of Chap. I. For the adapted frame (X, c~) defined in sec. 1 we have

(1) a ~ = g~l'dxt.

We put p~ ~-- 8~z/Sx~ 8xj and note tha t

g - ~ ( X - - ~ x,e,) = (0, ..., O, 1) .

5Tow

Hence,

and

(2)

o r

(3)

dc~-= d(g -11~) e~ + g-1/~(0, ..., 0, 1) ~ p~ dx~

= g -~ - l l~ (~p~dx j )X -~ terms in cl, ..., c . .

* :- g-(n+2)tn a~ ~ pij (r ~

a*A...Aa* = g-(~+~)det//(z)MA,..Aa ~ .

A ---- g-~+2) det H(z).

The differential a iA. . .Aa ", is (n~-1) times the element of volume of a cone of apex 0; the volume of K is the integral of the volume of these cones. The volume

H. GUGGE:~i~a~ER: Applications o] polarity 379

of K* is the sum of the integral over ( - -1)"Aa~A. . .Aa" and the volume of the convex hull of 0 and the cylindrical surface over aG*. Le t h(co) be the suppor t func- t ion of G in E" evaluated for the uni t vectors co E S~-h We also denote the surface e lement of S ~-~ by e/co. The polar equat ion of aG* is r----h-~(co). Then we obta in:

PI~OPOSI~rION 4. - I] z satis]ies the eone/itions of the theorem then,

(7 ~7

The formula can be simplified since for convex ( / a n d vanishing boundary values,

ge/xl ... dx~ -~ (n-~ 1) f zdx l ... dx~. (~

4. - The proposit ions of the last sections show that , for the inequalities of the theorem the generalization of the second der ivat ive is the Hessian detH(z) or some funct ion connected with this determinant . The mat r ix H(z) appears in several other respects as generaliza~tion of the second derivat ive. For example, in a recent query [13], S. ZAID~N asks for generalizations to n variables of the inequal i ty

f¢l

m a x ty(x)t ly"(x)l x ~0

which is valid and sharp for C -~ functions vanishing at Xo and xl. The nex t proposi- t ion gives a similar inequal i ty where Iyt'i is replaced b y the operator norm l]H(z)[[, the absolute largest eigenvalue of the symmetr ic mat r ix H(z).

P~OI'OSn:ION 5. - Zet K be a convex domain in E ~ ane/ z a C 2 ]unction in K that vanishes on ~K. Zet ~(x, co) be the length o] the chore/ c(x, co) o] K through x ~ K in the direction o] the unit vector co ~ S ~-1 and denote the arclength an c(x, o~) by s. Then

Let xo e K be a point where Iz] is maximal . We use polar coordinates (r, co) of center xo. The equation of ~K is r = r(co). Then

z(xo) = fzre/r = f[r(co)-rJz.e/r 0 0

380 H. GUGGENHEIM:ER: Applications o] polarity

o r

J . e , ,

(4)

,.o)1 .(o)iti (o)II ,. 0

,j max lz(x)l< max min r(w 11~(~)11 d~ ~ K ~ K a ~ a - I

0

from which the (in general, weaker) inequality of the proposition is derived im- mediately.

5. - The appearance of the expression v = z - - ~ x ~ p ~ might suggest a connec- tion with the Legendre transformation, i.e., the contact transformation derived from polarity with respect to the paraboloid of revolution 2z = ~x~. However~ the I~egendre transformation transforms arcs without inflections and cusps into ares of the same type but does not preserve concavity towards the origin. For exampl% the transformation of the unoriented line elements in the plane

X = y' :Y = x y ' - - y

maps any arc y = y(x) that satisfies the hypotheses of prop. 1 onto an are convex towards the X-axis. In this case, for Xo,V,O<xl,

f dx(x) = f (xy'-,):,"

> ½[y'(xo) + ly'(~l) il Ixoy'(xo) + ~y'(x,)l .

H I . - P o l a r i t y a s c o n t a c t t r a n s f o r m a t i o n .

1. - The polarity in the unit sphere

X~ + . . . -{'- x~ + z ~ = 1

induces a transformation of surface elements

(x, z; p) dz = ~.,Io~dx~ = pT dx

into elements

(x*, z; p*) dz* = p *~ dx*

by (1), I I :

(1) z * = g-1 x~ '=- -p~g-1 .

H. GUGGENn~,~E~: Appl icat ions o] polarity 381

Here we use m a t r i x notat ion, x = (x~), p = (p~) as row vectors (and the same for differentials), x ~ @ p = (x~pj), I : (~j), H : (P~s). Since polar i ty is an involution,

i t follows also t h a t x~------/J~ y ------p~ z and

(2) p* = - x , z -1 .

We can compute the ma t r ix H* -~ (a2z*/~x * ~x*) f rom dp =- H d x , dp* -= H * d x * and

dp* = - - d(xz -1) = z-~Cx r Q p - - z I ) d x .

Then also,

dp = z*-~(x*~ ® p* - - z* I ) ,ix*

= (zz*)-l(p ~ ® x - - zI) , ix*.

For nonsingtdar Hessians we obta in

dp* = z-~-(x ~ ® p - - z I ) H - l d p

= z-~z*-l(x ~" ® p - - z I ) H - 1 ( x ~ ® p - - zI ) ~ dx*

o r

( ) ; (3) H * = g- xT ® p I H -~ x ~ - p I . Z

For the de te rminan t s we obtain the easier fo rmula

d e t H * ---- (zz*)-" d e t H -~ det x ~--(5~

= (g/z).+~(detH) - ' ,

i . e ,

(4) z ~+~ det H(z) z *'+~- det H(z*) = 1 .

The last fo rmula also follows f rom ( 9 ) I and ( 3 ) I I .

2. - JORGEh'S, CALABI and POGO~ELOV [2, 7, 9] p roved the following theorem: A complete , convex C 5 surface in E ~+~ which is schlicht over the whole p lane z ~ 0 and a solution of d e t H ( z ) = 1 is an elliptic paraboloid. Geometrical ly, this is an example of ~ surface whose equat ion is inva r i an t in the Legendre t r ans fo rmat ion

(5 ) X ---- p , P ---- x , Z ---- - - g

25 - l n n a l i d t M a t e m a t i c a

382 H. GUC~E~m~n~E~: Application, s of polarity

and which under suitable conditions of global regulari ty admits only surfaces tha t form a family which is invariant under the Legendrc transformation. In order to t ransform this theorem by polari ty in the uni t sphere we add the condition tha t the surface z----z(x) should be an affine surface extendable to a projective nonsingular surface. Then its polar image is an elliptic hypersurface intersecting any parallel to the z-axis at most two times, without double tangent planes and with at most one point in z = 0. Hence, we obtain from (4):

Pl~OPOSITIO~ 1. - A sufficiently di//erentiable, elliptic surface given by an at most bivalent solution of

det tt(z) = (1 -- z -1 Z x~P') "+~

that is the af]ine manifold o / a complete compact projective hypersurface without double tangent planes with at most one point on z = 0 is a quadric through 0 that has the z-axis as one of its axes.

The Gauss curvature K of a surface z = z(x) is given by

K = detH(z). (1 -~ Zp~) -(~+2)/2

(ef. (7), Chap. I., and (3), Chap. II) . We introduce t X l = ( x ~ - . . . - ~ x ~ - z 2 ) ½. From (2) and (4) we obtain

(6) K* = g~+2 lXl ~+~ detH(z)

Since, also, h = IX*I -~ is the support distance, we have K = ( X * g ) - ~ - ~ d e t H ( z ) by (7), Chap. I and (3), Chap I I and obtain a parallel to (4):

P]~oPos:[~rio~ 2. - The Gauss curvatures K(z) and K(z*) o / a surface z-----z(x) and its polar in the unit sphere are connected by

]Xl~+~K(z) IX* ]~+~K(z*) = ] .

We note tha t is similarly easy to compute the transformation of the Gauss cur- vature in a Legendre transformation since, there, det H(Z) ---- detH(z) -1. Hence, for the corresponding curvatures we have

(7) K(z)K(Z) ---- [(1 + ~p~)(1 -[- ~x~)] -(,,+e)/2

Since every complete surface of constant curvature is a sphere, we obtain:

PI~oPosI~I0~ 3. - A smooth hypersurface that is the affine manifold of a complete compact projective hypersur/ace and is given by a solution either o/

[ g ~+~ detH(z) = \~-~]

H. G v ~ E ~ m ~ n ~ : Applications o] polarity 383

or ot

det H(z ) -~ (1-~ ~x~) -("+2)/~

is a quadric.

Similarly, the t ransform, b y (3), of the equat ion of minimal sttrf~ces is

r ( (yq- -z ) ~ ~- x'-(1 ~- q~)} ~- t((xp - - z ) ~ ~- y~(1 ~- p2)} ~_

~- 2s{z(xq ~- yp) ~- x y - - (x ~ + y'-)pq} = O .

B y the Berns te in theorem, u solution of this equat ion t h a t b y polar i ty t ransforms

into a surface schlicht over z-= 0 is a constant .

REFERENCES

[I] R. P. BA~'~BA~, Polar reciprocal convex bodies, Proe. Cambridge Phil. Soc., 51 (1955), pp. 377-378.

[2] E. C).LABX, Improper amine hyperspheres o] convex type and a generalization o] a theorem of K. J6rgens, ~ichigan J. M~th., 5 (1958), pp. 105-126.

[3] I-L GVGGI~NHEI~R, Dif]erentiat Geometry, McGraw-Hill, New York, 1963. [4] H. GVGG~NH~IME~, Hill equations with coexisting periodic solutions, J. DiD. Equ., 5

(1969), pp. 159-166. [5] It. GUGGENHEIM~, Polar reciprocal convex bodies, Israel J. Math., 44 (1973), pp. 309-316. [6] H. GVOGE~H~IMER, ~ber das Verhalten der Gausschen Kri~mmung bei A]]initat, El. Math.,

28 (1973), p. 42. [7] K. JS~Gv.NS, ~ber die Lvsungen der Dif/erentialgleichung rt 2 ~ s ~ --: 1, Math. Ann., 427

(1954), pp. 130-134. [8] D. L~UGWITZ, Di]]erentialgcometrie in Vektorrdumen, Vieweg, Braunschweig, 1965. [9] A. V. POGOn~r.OV, On the improper convex af]ine hyperspheves, Geometriue dedicata, 1

(1972), pp. 33-46. [10] E. S~LKOWSKI, A//ine Dif]erentiaIgeometrie, Teubner, Leipzig-Berlin, 1929. [11] P. A. und A. P. Sc~I~o~:ow, A/]ine DiMerentialgeametrie, Teubner, Leipzig, 1962. [12] E. J. WIT,CZYNSKI, Projective diMerential geometry o] curves and ruled sur]aces, Teubner,

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