H(f · g0) = (a’1) tAH(f)A,

where g0 ∈ G0 is as in (2.26).6 Now we consider the quantity

def

∆(f) = det(Hij (f)),

which vanishes at some point of F0 (M ) if and only if it vanishes on the entire

¬ber containing that point. We will say that M ‚ An+1 is non-degenerate if

∆ = 0 everywhere on F0 (M ). Also note that the absolute signature of Hij is

well-de¬ned at each point of M . It is easy to see that Hij is de¬nite if and

only if M ‚ An+1 is convex. In what follows, we will assume that M is a

non-degenerate hypersurface, but not necessarily that it is convex.

It turns out that T = (Tijk ), which one would like to regard as a sort of

covariant derivative of H = (Hij ), is not a tensor; that is, it does not transform

by a linear representation along the ¬bers of F0(M ) ’ M . We will exploit this

below to reduce the principal bundle F0 (M ) ’ M to a subbundle of frames

satisfying a higher-order adaptivity condition. Namely, F1(M ) ‚ F0(M ) will

consist of those frames where Tijk is traceless with respect to the non-degenerate

symmetric bilinear form Hij , meaning H jk Tijk = 0, where (H ij ) is the matrix

inverse of (Hij ). Geometrically, the reduction will amount to a canonical choice

of line ¬eld Re0 transverse to M , which we will think of as giving at each point

of M a canonical a¬ne normal line.

To justify this, we let (H ij ) denote the matrix inverse of (Hij ), and let

def

Ci = H jk Tijk

be the vector of traces of T with respect to H. We compute

∆’1d∆

d(log ∆) =

Tr(H ’1 dH)

=

H ij dHij

=

’n•0 + 2•i + Ciωi .

= 0 i

6 As

usual, our argument only proves this claim for g0 in the identity component of G0 ,

but it may be checked directly for representative elements of each of the other components.

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES 61

Now di¬erentiate again and collect terms to ¬nd

0 = (dCi ’ Cj •j ’ (n + 2)Hij •j ) § ωi . (2.27)

i 0

Therefore, we have

dCi ≡ Cj •j + (n + 2)Hij •j (mod {ω1 , . . . , ωn }), (2.28)

0

i

which expresses how the traces Ci vary along the ¬bers of F0 (M ) ’ M . In

particular, if the matrix (Hij ) is non-singular, as we are assuming, then the

action of the structure group on the values of the vector (Ci ) ∈ Rn is transitive;

that is, every value in Rn is taken by (Ci) in each ¬ber. Therefore, the set of

0-adapted frames f ∈ F0 (M ) where each Ci (f) = 0 is a principal subbundle

F1 (M ) ‚ F0 (M ), whose structure group is the stabilizer of 0 ∈ Rn under the

action. This stabilizer is

a0

def

: a ∈ R— , A ∈ GL(n, R) .

G1 = g1 =

0A

Comparing to the full action (2.24) of the a¬ne group A(n + 1) on F, we see

that along each ¬ber of F1(M ), the direction Re0 is ¬xed. Thus, we have

uniquely chosen the direction of e0 at each point of M by the condition Ci = 0

for i = 1, . . . , n.

A more concrete explanation of what we have done is seen by locally pre-

senting our hypersurface in the form

1¯ 1¯

x0 = Hij (0)xixj + Tijk (x1, . . . , xn)xixj xk .

2 6

An a¬ne change of coordinates that will preserve this form is the addition of a

multiple of x0 to each xi; the n choices that this entails can be uniquely made

¯ ¯

so that Tijk(0) is traceless with respect to Hij (0). Once such choices are ¬xed,

‚

then so is the direction of ‚x0 , and this gives the canonical a¬ne normal line at

x = 0.

There is a remarkable interpretation of the a¬ne normal direction at a point

where Hij is positive-de¬nite (see [Bla67]). Consider the 1-parameter family of

hyperplanes parallel to the tangent plane at the given point. For those planes

su¬ciently near the tangent plane, the intersection with a ¬xed neighborhood

in the surface is a closed submanifold of dimension n ’ 2 in M , having an a¬ne-

invariant center-of-mass. These centers-of-mass form a curve in a¬ne space,

passing through the point of interest; this curve™s tangent line at that point is

the a¬ne normal direction.

We can see from (2.28) that on F1 (M ), where Tijk is traceless, the forms •j 0

are semibasic over M . It is less convenient to express these in terms of the basis

ωi than to instead use •0 = Hij ωj , assuming that M ‚ An+1 is non-degenerate.

i

On F1 (M ) we write

•j = U jk •0 .

0 k

Now (2.27), restricted to F1 (M ) where Ci = 0, implies that U ij = U ji.

´

62 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

The reader may carry out computations similar to those above to show that

the (U ij ) and (Tijk ) are tensors; that is, they transform along the ¬bers of

F1 (M ) by a linear representation of G1. For example, if Tijk = 0 at some point

of F1(M ), then Tijk = 0 everywhere along the same ¬ber of F1 (M ) ’ M .

Furthermore, the transformation law for U ij is such that if U ij = »H ij at some

point, for some », then the same is true”with possibly varying »”everywhere

on the same ¬ber. We can now give some additional interpretations of the

simplest cases of the a¬ne second fundamental form U ij and the a¬ne cubic

form Tijk. The following theorem is the main purpose of this digression.

Theorem 2.3 (1) If U ij = »H ij everywhere on F1 (M )”that is, if the second

fundamental form is a scalar multiple of the ¬rst fundamental form”then either

» = 0 everywhere or » = 0 everywhere. In the ¬rst case, the a¬ne normal lines

of M are all parallel, and in the second case, the a¬ne normal lines of M are

all concurrent.

(2) If Tijk = 0 everywhere on F1(M ), then U ij = »H ij everywhere. In this

case, if » = 0, then M is a paraboloid, while if » = 0, then M is a non-degenerate

quadric.

Proof. Suppose ¬rst that U ij = »H ij on F1 (M ) for some function ». This is

same as writing

•j = »H jl •0 = »ωj .

l

0

We di¬erentiate this equation (substituting itself), and obtain

(d» ’ »•0 ) § ωj = 0 for each j.

0

Under the standing assumption n > 1, this means that

d» = »•0 .

0

So assuming that M is connected, we have the ¬rst statement of (1). We will

describe the geometric consequences of each of the two possibilities.

First, suppose that » = 0, so that U ij = 0, and then •j = 0 throughout

0

F1 (M ). Then the de¬nition of our original basis of 1-forms gives

de0 = ea •a = e0 •0 ,

0 0

meaning that the direction in An+1 of e0 is ¬xed throughout F1 (M ), or equiv-

alently, all of the a¬ne normals of M are parallel.

Next, suppose » = 0, and assume for simplicity that » < 0. The di¬erential

equation d» = »•0 implies that we can restrict to the principal subbundle

0

F2 (M ) where » = ’1. This amounts to a choice of a particular vector ¬eld e0

along the a¬ne normal line ¬eld already de¬ned. Note that on F2 (M ), we have

•j = ’ωj = ’H jk •0 , •0 = 0. (2.29)

k 0

0

As a result, the structure equations dx = ei ωi and de0 = ea •a = ’ei ωi imply

0

that

d(x + e0 ) = 0,

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES 63

so that x + e0 is a constant element of An+1. In particular, all of the a¬ne

normal lines of M pass through this point. This completes the proof of (1).

Now assume that Tijk = 0 identically; this will be satis¬ed by each member

of the ¬elds of a¬ne hypersurfaces associated to certain neo-classical Poincar´-

e

ij ij

Cartan forms of interest. Our ¬rst claim is that U = »H for some function

» on F1 (M ). To see this, note that our hypothesis means

dHij = ’Hij •0 + Hkj •k + Hik •k .

0 i j

We di¬erentiate this, using the structure equations in the simpli¬ed form that

de¬ned the reduction to F1 (M ), and obtain

0 = ’Hkj •k § •0 ’ Hik •k § •0 .

0 i 0 j

If we use Hij to raise and lower indices and de¬ne

Uij = Hik Hlj U kl ,

then the preceding equation may be written as

0 = ’(Ujl Hik + Uil Hjk)ωl § ωk .

The coe¬cients of this vanishing 2-form then satisfy

0 = Ujl Hik + Uil Hjk ’ Ujk Hil ’ Uik Hjl ;

we multiply by H ik (and sum over i, k) to conclude

1 ik

Ujl = (H Uik )Hjl .

n

This proves that

U ij = »H ij ,

1

with » = n H kl Ukl .

We now return to the possibilities » = 0, » = 0 under the stronger hypothesis

Tijk = 0.

In the ¬rst case, note that with the condition •j = 0 on F1(M ), we have

0

0 i

that the Pfa¬an system generated by •0 and •j (for 1 ¤ i, j ¤ n) is integrable.

˜ ˜

Let M be any leaf of this system. Restricted to M , we have

dHij = 0,

so that the functions Hij are constants. Furthermore, the linearly independent

˜

1-forms ωi on M are each closed, so that (at least locally, or else on a simply

˜

connected cover) there are coordinates ui on M with

ωi = dui.

Substituting all of this into the structure equations, we have:

´

64 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

˜

• de0 = 0, so that e0 is a constant element of An+1 on M;

• dei = e0 •0 = e0 Hij ωj = d(e0 Hij uj ), so that

i

ei = ei + e0 Hij uj

¯

for some constant ei ∈ An+1 ;

¯

1

• dx = ei ωi = (¯i + e0 Hij uj )dui = d(ui ei + 2 e0 Hij uiuj ), so that

e ¯

1

x = x + uiei + Hij uiuj e0

¯ ¯

2

for some constant x ∈ An+1.

¯

˜

The conclusion is that as the coordinates ui vary on M, the An+1 -valued func-

˜

tion x on M traces out a paraboloid, with vertex at x and axis along the direction

¯

of e0 .

Turning to the case » = 0, recall that under the assumption » < 0, we

can reduce to a subbundle F2 (M ) ‚ F1 (M ) on which » = ’1. We use the

di¬erential equation

dHij = Hik •k + Hkj •k

j i

¯

to reduce again to a subbundle F3 (M ) ‚ F2 (M ) on which Hij = Hij is some

i

constant matrix. On F3 (M ), the forms •j satisfy linear algebraic relations

¯ ¯

0 = Hik •k + Hkj •k .

j i

Our assumption » = ’1 allows us to combine these with the relations (2.29) by

de¬ning

0 •0 10

j

¦= , H= ,

¯

i i

•0 •j 0H

and then

H¦ + t ¦H = 0.

In other words, the matrix-valued 1-form ¦ on F3 (M ) takes values in the Lie

algebra of the stabilizer of the bilinear form H. For instance, if our hyper-

surface M is convex, so that (Hij ) is de¬nite everywhere, then we could have

¯

chosen Hij = δij , and then ¦ would take values in the Lie algebra so(n + 1, R).

Whatever the signature of Hij , let the stabilizer of H be denoted by O(H) ‚