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This M will be given the structure of a contact manifold in such a way that
transverse Legendre submanifolds correspond to arbitrary immersed hypersur-
faces in En+1 . Note that M may be identiп¬Ѓed with the unit sphere bundle of
En+1 by associating to a contact element (x, H) its oriented orthogonal com-
plement (x, e0 ). We will use this identiп¬Ѓcation without further comment.
The projection F в†’ M taking (x, (ea )) в†’ (x, e0) is E(n + 1, R)-equivariant
(for the left-action). To describe the contact structure on M and to carry out
calculations, we will actually work on F using the following structure equations.
First, we deп¬Ѓne canonical 1-forms on F by diп¬Ђerentiating the vector-valued
coordinate functions x(f), ea (f) on F, and decomposing the resulting vector-
valued 1-forms at each f в€€ F with respect to the frame ea (f):

eb В· П‰ b , b
e b В· П‰a .
dx = dea = (1.16)

Diп¬Ђerentiating the relations ea (f), eb (f) = Оґab yields
a b
П‰b + П‰a = 0.

The forms П‰a , П‰b satisfy no other linear algebraic relations, giving a total of
a
1
(n + 1) + 2 n(n + 1) = dim(F) independent 1-forms. By taking the derivatives
of the deп¬Ѓning relations (1.16), we obtain the structure equations

dП‰a + П‰c в€§ П‰c = 0,
a a a c
П‰c в€§ П‰b = 0.
dП‰b + (1.17)

The forms П‰a are identiп¬Ѓed with the usual tautological 1-forms on the orthonor-
mal frame bundle of a Riemannian manifold (in this case, of En+1 ); and then the
a
п¬Ѓrst equation indicates that П‰b are components of the Levi-Civita connection
of En+1 , while the second indicates that it has vanishing Riemann curvature
tensor.
In terms of these forms, the п¬Ѓbers of x : F в†’ En+1 are exactly the maximal
connected integral manifolds of the Pfaп¬ѓan system {П‰ a }. Note that {П‰ a } and
{dxa } are alternative bases for the space of forms on F that are semibasic over
En+1 , but the former is E(n + 1)-invariant, while the latter is not.
We return to an explanation of our contact manifold M , by п¬Ѓrst distinguish-
ing the 1-form on F
def
Оё = П‰0 .
Note that its deп¬Ѓning formula

v в€€ Tf F,
Оёf (v) = dx(v), e0(f) ,

shows that it is the pullback of a unique, globally deп¬Ѓned 1-form on M , which
we will also call Оё в€€ в„¦1 (M ). To see that Оё is a contact form, п¬Ѓrst relabel the
forms on F (this will be useful later, as well)

def
0
ПЂi = П‰ i ,
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 23

and note the equation on F

ПЂi в€§ П‰ i .
dОё = в€’

So on F we certainly have Оё в€§ (dОё)n = 0, and because pullback of forms via the
submersion F в†’ M is injective, the same non-degeneracy holds on M .
To understand the Legendre submanifolds of M , consider an oriented im-
mersion
О№
N n в†’ M 2n+1, y = (y1 , . . . , yn) в†’ (x(y), e0 (y)).
The Legendre condition is

(О№в€— Оё)y (v) = dxy (v), e0 (y) = 0, v в€€ Ty N.

In the transverse case, when the composition x в—¦ О№ : N n в†’ M 2n+1 в†’ En+1 is a
hypersurface immersion (equivalently, О№в€— ( П‰i ) = 0, suitably interpreted), this
condition is that e0 (y) is a unit normal vector to the hypersurface xв—¦О№(N ). These
Legendre submanifolds may therefore be thought of as the graphs of Gauss maps
of oriented hypersurfaces N n в†’ En+1 . Non-transverse Legendre submanifolds
of M are sometimes of interest. To give some intuition for these, we exhibit
two examples in the contact manifold over E3 . First, over an immersed curve
x : I в†’ E3 , one can deп¬Ѓne a cylinder N = S 1 Г— I в†’ M в€ј E3 Г— S 2 by
=

(v, w) в†’ (x(w), Rv (ОЅx)),

where ОЅ is any normal vector п¬Ѓeld along the curve x(w), and Rv is rotation
through angle v в€€ S 1 about the tangent x (w). The image is just the unit
normal bundle of the curve, and it is easily veriп¬Ѓed that this is a Legendre
submanifold.
Our second example corresponds to the pseudosphere, a singular surface
of revolution in E3 having constant Gauss curvature K = в€’1 away from the
singular locus. The map x : S 1 Г— R в†’ E3 given by

x : (v, w) в†’ (sech w cos v, в€’sech w sin v, w в€’ tanh w)

fails to be an immersion where w = 0. However, the Gauss map of the comple-
ment of this singular locus can be extended to a smooth map e3 : S 1 Г— R в†’ S 2
given by
e3(v, w) = (в€’tanh w cos v, tanh w sin v, в€’sech w).
The graph of the Gauss map is the product (x, e3) : S 1 Г— R в†’ M . It is a
Legendre submanifold, giving a smooth surface in M whose projection to E3
is one-to-one, is an immersion almost everywhere, and has image equal to the
singular pseudosphere. We will discuss in В§4.3.3 the exterior diп¬Ђerential system
whose integral manifolds are graphs of Gauss maps of K = в€’1 surfaces in E3 .
In В§4.3.4, we will discuss the BВЁcklund transformation for this system, which
a
relates this particular example to a special case of the preceding example, the
unit normal bundle of a line.
Вґ
24 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

1.4.2 Euclidean-invariant Euler-Lagrange Systems
We can now introduce one of the most important of all variational problems,
that of п¬Ѓnding minimal-area hypersurfaces in Euclidean space. Deп¬Ѓne the n-
form
О› = П‰1 в€§ В· В· В· в€§ П‰n в€€ в„¦n (F),
and observe that it is basic over M ; that is, it is the pullback of a well-deп¬Ѓned
n-form on M (although its factors П‰ i are not basic). This deп¬Ѓnes a Lagrangian
functional
FО› (N ) = О›
N

on compact Legendre submanifolds N n в†’ M 2n+1, which in the transverse case
discussed earlier equals the area of N induced by the immersion N в†’ En+1 .
We calculate the PoincarВґ-Cartan form up on F using the structure equations
e
(1.17), as
dО› = в€’Оё в€§ ПЂi в€§ П‰(i),
so the Euler-Lagrange system EО› is generated by I = {Оё, dОё} and

ОЁ=в€’ ПЂi в€§ П‰(i),

which is again well-deп¬Ѓned on M . A transverse Legendre submanifold N в†’
M will locally have a basis of 1-forms given by pullbacks (by any section) of
П‰1 , . . . , П‰n, so applying the Cartan lemma to

0 = dОё|N = в€’ПЂi в€§ П‰i

shows that restricted to N there are expressions

hij П‰j
ПЂi =
j

for some functions hij = hji. If N в†’ M is also an integral manifold of EО› вЉ‚
в„¦в€— (M ), then additionally

hii П‰1 в€§ В· В· В· в€§ П‰n.
0 = ОЁ|N = в€’

One can identify hij with the second fundamental form of N в†’ En+1 in this
transverse case, and we then have the usual criterion that a hypersurface is sta-
tionary for the area functional if and only if its mean curvature hii vanishes.
We will return to the study of this Euler-Lagrange system shortly.
Another natural E(n+1)-invariant PDE for hypersurfaces in Euclidean space
is that of prescribed constant mean curvature H, not necessarily zero. We п¬Ѓrst
ask whether such an equation is even Euler-Lagrange, and to answer this we
apply our inverse problem test to the Monge-Ampere system

EH = {Оё, dОё, ОЁH }, ОЁH = в€’ ПЂi в€§ П‰(i) в€’ HП‰ .
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 25

Here, H is the prescribed constant and П‰ = П‰ 1 в€§ В· В· В· в€§ П‰n is the induced vol-
ume form. The transverse integral manifolds of EH correspond to the desired
Euclidean hypersurfaces.
To implement the test, we take the candidate PoincarВґ-Cartan form
e

О H = в€’Оё в€§ ПЂi в€§ П‰(i) в€’ HП‰

and diп¬Ђerentiate; the derivative of the п¬Ѓrst term vanishes, as we know from the
preceding case of H = 0, and we have

H d(Оё в€§ П‰1 в€§ В· В· В· в€§ П‰n )
dО H =
H d(dx0 в€§ В· В· В· в€§ dxn)
=
= 0.

So this EH is at least locally the Euler-Lagrange system for some functional О›H ,
which can be taken to be an anti-derivative of О H . One diп¬ѓculty in п¬Ѓnding О H
is that there is no such О›H that is invariant under the Euclidean group E(n+1).
The next best thing would be to п¬Ѓnd a О›H which is invariant under the rotation
subgroup SO(n + 1, R), but not under translations. A little experimentation
yields the Lagrangian
H
О›H = П‰ + n+1 x в„¦, dО›H = О H ,
в€‚
xa в€‚xa is the radial position vector п¬Ѓeld, П‰ = П‰ 1 в€§ В· В· В· в€§ П‰n is the
where x =
hypersurface area form, and в„¦ = П‰ 1 в€§ В· В· В· в€§ П‰n+1 is the ambient volume form.
The choice of an origin from which to deп¬Ѓne the position vector x reduces the
symmetry group of О›H from E(n + 1) to SO(n + 1, R). The functional N О›H
gives the area of the hypersurface N plus a scalar multiple of the signed volume
of the cone on N with vertex at the origin.
It is actually possible to list all of the Euclidean-invariant PoincarВґ-Cartan
e
n+1
forms on M в†’ E . Let
1
О›в€’1 = в€’ n+1 x ПЂI в€§ П‰(I) (0 в‰¤ k в‰¤ n),
в„¦, О›k =
|I|=k

and
О k = в€’Оё в€§ О›k ,
It is an exercise using the structure equations to show that

dО›k = О k+1.

Although these forms are initially deп¬Ѓned up on F, it is easily veriп¬Ѓed that
they are pull-backs of forms on M , which we denote by the same name. It
can be proved using the п¬Ѓrst fundamental theorem of orthogonal invariants
that any Euclidean-invariant PoincarВґ-Cartan form is a linear combination of
e
О 0 , . . . , О n. Note that such a PoincarВґ-Cartan form is induced by a Euclidean-
e
invariant functional if and only if О 0 is not involved.
Вґ
26 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

We can geometrically interpret О›k |N for transverse Legendre submanifolds N
as the sum of the k Г—k minor determinants of the second fundamental form IIN ,
times the hypersurface area form of N . In case k = n we have dО›n = О n+1 = 0,
reп¬‚ecting the fact that the functional

О›n = K dA
N N

is variationally trivial, where K is the Gauss-Kronecker curvature.

Contact Equivalence of Linear Weingarten Equations for Surfaces
The Euclidean-invariant PoincarВґ-Cartan forms for surfaces in E3 give rise to
e
the linear Weingarten equations, of the form
aK + bH + c = 0
for constants a, b, c. Although these second-order PDEs are inequivalent under
point-transformations for non-proportional choices of a, b, c, we will show that
under contact transformations there are only п¬Ѓve distinct equivalence classes of
linear Weingarten equations.
To study surfaces, we work on the unit sphere bundle ПЂ : M 5 в†’ E3 , and
recall the formula for the contact form
v в€€ T(x,e0 ) M.
Оё(x,e0 ) (v) = ПЂв€— (v), e0 ,
We deп¬Ѓne two 1-parameter groups of diп¬Ђeomorphisms of M as follows:
П•t(x, e0 ) = (x + te0 , e0),
П€s(x, e0 ) = (exp(s)x, e0 ).
It is not hard to see geometrically that these deп¬Ѓne contact transformations on
M , although this result will also come out of the following calculations. We
will carry out calculations on the full Euclidean frame bundle F в†’ E3 , where
there is a basis of 1-forms П‰ 1 , П‰2 , Оё, ПЂ1 , ПЂ2, П‰2 satisfying structure equations
1

presented earlier.
в€‚
To study П•t we use its generating vector п¬Ѓeld v = в€‚Оё , which is the dual of
the 1-form Оё with respect to the preceding basis. We can easily compute Lie
derivatives
Lv П‰1 = в€’ПЂ1 , Lv П‰2 = в€’ПЂ2 , Lv Оё = 0, Lv ПЂ1 = 0, Lv ПЂ2 = 0.
Now, the п¬Ѓbers of F в†’ M have tangent spaces given by {П‰ 1, П‰2 , Оё, ПЂ1, ПЂ2}вЉҐ ,
and this distribution is evidently preserved by the п¬‚ow along v. This implies
that v induces a vector п¬Ѓeld downstairs on M , whose п¬‚ow is easily seen to be
П•t . The fact that Lv Оё = 0 conп¬Ѓrms that П•t is a contact transformation.
We can now examine the eп¬Ђect of П•t on the invariant Euler-Lagrange systems
corresponding to linear Weingarten equations by introducing
ОЁ1 = ПЂ1 в€§ П‰ 2 в€’ ПЂ2 в€§ П‰ 1 , ОЁ0 = П‰ 1 в€§ П‰ 2 .
ОЁ2 = ПЂ 1 в€§ ПЂ2 ,
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 27
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