<<

. 7
( 48 .)



>>


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 identi¬ed 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 identi¬cation without further comment.
The projection F ’ M taking (x, (ea )) ’ (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 de¬ne canonical 1-forms on F by di¬erentiating the vector-valued
coordinate functions x(f), ea (f) on F, and decomposing the resulting vector-
valued 1-forms at each f ∈ F with respect to the frame ea (f):

eb · ω b , b
e b · ωa .
dx = dea = (1.16)

Di¬erentiating the relations ea (f), eb (f) = δab yields
a b
ωb + ωa = 0.

The forms ωa , ωb 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 de¬ning relations (1.16), we obtain the structure equations

dωa + ωc § ωc = 0,
a a a c
ωc § ωb = 0.
dωb + (1.17)

The forms ωa are identi¬ed 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
¬rst equation indicates that ωb 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 ¬bers of x : F ’ En+1 are exactly the maximal
connected integral manifolds of the Pfa¬an system {ω a }. Note that {ω 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 ¬rst distinguish-
ing the 1-form on F
def
θ = ω0 .
Note that its de¬ning formula

v ∈ Tf F,
θf (v) = dx(v), e0(f) ,

shows that it is the pullback of a unique, globally de¬ned 1-form on M , which
we will also call θ ∈ „¦1 (M ). To see that θ is a contact form, ¬rst relabel the
forms on F (this will be useful later, as well)

def
0
πi = ω i ,
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 23

and note the equation on F

πi § ω i .
dθ = ’

So on F we certainly have θ § (dθ)n = 0, and because pullback of forms via the
submersion F ’ M is injective, the same non-degeneracy holds on M .
To understand the Legendre submanifolds of M , consider an oriented im-
mersion
ι
N n ’ M 2n+1, y = (y1 , . . . , yn) ’ (x(y), e0 (y)).
The Legendre condition is

(ι— θ)y (v) = dxy (v), e0 (y) = 0, v ∈ Ty N.

In the transverse case, when the composition x —¦ ι : N n ’ M 2n+1 ’ En+1 is a
hypersurface immersion (equivalently, ι— ( ωi ) = 0, suitably interpreted), this
condition is that e0 (y) is a unit normal vector to the hypersurface x—¦ι(N ). These
Legendre submanifolds may therefore be thought of as the graphs of Gauss maps
of oriented hypersurfaces N n ’ 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 ’ E3 , one can de¬ne a cylinder N = S 1 — I ’ M ∼ E3 — S 2 by
=

(v, w) ’ (x(w), Rv (νx)),

where ν is any normal vector ¬eld along the curve x(w), and Rv is rotation
through angle v ∈ S 1 about the tangent x (w). The image is just the unit
normal bundle of the curve, and it is easily veri¬ed 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 = ’1 away from the
singular locus. The map x : S 1 — R ’ E3 given by

x : (v, w) ’ (sech w cos v, ’sech w sin v, w ’ 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 — R ’ S 2
given by
e3(v, w) = (’tanh w cos v, tanh w sin v, ’sech w).
The graph of the Gauss map is the product (x, e3) : S 1 — R ’ 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 §4.3.3 the exterior di¬erential system
whose integral manifolds are graphs of Gauss maps of K = ’1 surfaces in E3 .
In §4.3.4, we will discuss the B¨cklund 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.
´
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 ¬nding minimal-area hypersurfaces in Euclidean space. De¬ne the n-
form
Λ = ω1 § · · · § ωn ∈ „¦n (F),
and observe that it is basic over M ; that is, it is the pullback of a well-de¬ned
n-form on M (although its factors ω i are not basic). This de¬nes a Lagrangian
functional
FΛ (N ) = Λ
N

on compact Legendre submanifolds N n ’ M 2n+1, which in the transverse case
discussed earlier equals the area of N induced by the immersion N ’ En+1 .
We calculate the Poincar´-Cartan form up on F using the structure equations
e
(1.17), as
dΛ = ’θ § πi § ω(i),
so the Euler-Lagrange system EΛ is generated by I = {θ, dθ} and

Ψ=’ πi § ω(i),

which is again well-de¬ned on M . A transverse Legendre submanifold N ’
M will locally have a basis of 1-forms given by pullbacks (by any section) of
ω1 , . . . , ωn, so applying the Cartan lemma to

0 = dθ|N = ’πi § ωi

shows that restricted to N there are expressions

hij ωj
πi =
j

for some functions hij = hji. If N ’ M is also an integral manifold of EΛ ‚
„¦— (M ), then additionally

hii ω1 § · · · § ωn.
0 = Ψ|N = ’

One can identify hij with the second fundamental form of N ’ 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 ¬rst
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 = {θ, dθ, ΨH }, ΨH = ’ πi § ω(i) ’ Hω .
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 25

Here, H is the prescribed constant and ω = ω 1 § · · · § ωn 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 Poincar´-Cartan form
e

ΠH = ’θ § πi § ω(i) ’ Hω

and di¬erentiate; the derivative of the ¬rst term vanishes, as we know from the
preceding case of H = 0, and we have

H d(θ § ω1 § · · · § ωn )
dΠH =
H d(dx0 § · · · § dxn)
=
= 0.

So this EH is at least locally the Euler-Lagrange system for some functional ΛH ,
which can be taken to be an anti-derivative of ΠH . One di¬culty in ¬nding ΠH
is that there is no such ΛH that is invariant under the Euclidean group E(n+1).
The next best thing would be to ¬nd a ΛH which is invariant under the rotation
subgroup SO(n + 1, R), but not under translations. A little experimentation
yields the Lagrangian
H
ΛH = ω + n+1 x „¦, dΛH = ΠH ,

xa ‚xa is the radial position vector ¬eld, ω = ω 1 § · · · § ωn is the
where x =
hypersurface area form, and „¦ = ω 1 § · · · § ωn+1 is the ambient volume form.
The choice of an origin from which to de¬ne the position vector x reduces the
symmetry group of ΛH from E(n + 1) to SO(n + 1, R). The functional N ΛH
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 Poincar´-Cartan
e
n+1
forms on M ’ E . Let
1
Λ’1 = ’ n+1 x πI § ω(I) (0 ¤ k ¤ n),
„¦, Λk =
|I|=k

and
Πk = ’θ § Λk ,
It is an exercise using the structure equations to show that

dΛk = Πk+1.

Although these forms are initially de¬ned up on F, it is easily veri¬ed that
they are pull-backs of forms on M , which we denote by the same name. It
can be proved using the ¬rst fundamental theorem of orthogonal invariants
that any Euclidean-invariant Poincar´-Cartan form is a linear combination of
e
Π0 , . . . , Πn. Note that such a Poincar´-Cartan form is induced by a Euclidean-
e
invariant functional if and only if Π0 is not involved.
´
26 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

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

Λn = 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 Poincar´-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 ¬ve distinct equivalence classes of
linear Weingarten equations.
To study surfaces, we work on the unit sphere bundle π : M 5 ’ E3 , and
recall the formula for the contact form
v ∈ T(x,e0 ) M.
θ(x,e0 ) (v) = π— (v), e0 ,
We de¬ne two 1-parameter groups of di¬eomorphisms of M as follows:
•t(x, e0 ) = (x + te0 , e0),
ψs(x, e0 ) = (exp(s)x, e0 ).
It is not hard to see geometrically that these de¬ne 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 ’ E3 , where
there is a basis of 1-forms ω 1 , ω2 , θ, π1 , π2, ω2 satisfying structure equations
1

presented earlier.

To study •t we use its generating vector ¬eld v = ‚θ , which is the dual of
the 1-form θ with respect to the preceding basis. We can easily compute Lie
derivatives
Lv ω1 = ’π1 , Lv ω2 = ’π2 , Lv θ = 0, Lv π1 = 0, Lv π2 = 0.
Now, the ¬bers of F ’ M have tangent spaces given by {ω 1, ω2 , θ, π1, π2}⊥ ,
and this distribution is evidently preserved by the ¬‚ow along v. This implies
that v induces a vector ¬eld downstairs on M , whose ¬‚ow is easily seen to be
•t . The fact that Lv θ = 0 con¬rms that •t is a contact transformation.
We can now examine the e¬ect of •t on the invariant Euler-Lagrange systems
corresponding to linear Weingarten equations by introducing
Ψ1 = π1 § ω 2 ’ π2 § ω 1 , Ψ0 = ω 1 § ω 2 .
Ψ2 = π 1 § π2 ,
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 27

<<

. 7
( 48 .)



>>