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Applications of contact geometry to first-order PDEs on manifolds
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Applications of contact geometry to first-order PDEs on manifolds
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Content
Applications of Contact Geometry to
First-Order PDEs on Manifolds
by
Alexander A. Sahakian
A Thesis Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
MASTER OF ARTS
(MATHEMATICS)
May 2018
Copyright 2018 Alexander A. Sahakian
Contents
Abstract i
Preface ii
1 Symplectic and Contact Geometry 1
1.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Contact Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 The 1-Jet Bundle . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 The Cosphere Bundle . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Relating the Contact Structures . . . . . . . . . . . . . . 19
2 Applications to First-Order PDEs 22
2.1 The Characteristic Field . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Construction of Solutions . . . . . . . . . . . . . . . . . . 25
2.2 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Huygens' Principle . . . . . . . . . . . . . . . . . . . . . . 31
References 33
Abstract
In this paper, we'll develop a method of geometrically solving rst-order
partial dierential equations (PDEs) of functions on manifolds. To do this, we'll
begin by considering the phase space (cotangent bundle) of a manifold, and the
natural, even-dimensional symplectic geometry associated with it. We'll then
\extend" these ideas to odd-dimensional manifolds and consider the analogous
theory of contact geometry that exists within them. In particular, we'll focus on
two important contact manifolds that arise in the analysis of PDEs, namely the
1-jet bundle J
1
B of a manifoldB representing physical space, and the cosphere
bundle PT
M of a manifoldM =BR representing space-time. We'll conclude
the paper by applying the theories we develop in these manifolds to actually
construct solutions of PDEs, and take a look at examples in the theory of wave
propagation.
i
Preface
In this paper, we'll extensively make use of manifolds and functions dened
on them. Unless otherwise stated, we'll assume by default that all manifolds
in question are smooth, and that all functions between our manifolds are also
smooth. We'll further assume that all of our manifolds are Riemannian; that is,
given a manifold M, we'll assume that it comes equipped with a Riemannian
metric that smoothly assigns to the tangent space corresponding to each point
z2M an inner product
h;i
z
: T
z
M T
z
M!R
and the associated norm
kk
z
: T
z
M!R
that give (T
z
M;h;i
z
) the structure of a Hilbert space. As such, we're able to
identify any tangent space T
z
M with its linear dual space, namely the cotan-
gent space T
z
M, by associating with each tangent vector v2 T
z
M the linear
functional ^ v2 T
z
M dened by
^ v() :=h;vi
z
:
We'll make use of this identication on multiple occasions. We're now ready to
begin with the content of the paper.
ii
Chapter 1
Symplectic and Contact Geometry
Suppose that we wish to consider the motion of a particle through time over an
n-dimensional manifoldB. In a rst-order system, simply knowing the particle's
instantaneous position gives us no information about where the particle was or
where it will go next. However, we can fully determine the motion of a particle
if we know its position in phase space, the direct sum of the position space and
the space of tangent directions, which may be thought of as momenta.
Locally, the position space B may be coordinatized by x = (x
1
;:::;x
n
),
and at each point x of B, the tangent space T
x
B consisting of all possible
momenta may be coordinatized by p = (p
1
;:::;p
n
), where p
j
corresponds to
the momentum in the x
j
-direction.
Moreover, we may think of a momentum p at x as a covector belonging to
the cotangent space at that point by identifying it with the linear functional
^ p2 T
x
B dened by
^ p() :=h;pi
x
:
As such, we see that the phase space of B is simply given by the cotangent
bundle T
B, with the 2n-dimensional coordinate system (x;p).
Figure 1.1: the phase space of a manifold
Certain conservation laws as well as other important results from classical
mechanics, typically given by purely algebraic restrictions, can be interpreted
as geometric constraints within this space. It's in this even-dimensional setting
that the theory of symplectic geometry arises; it allows us to study the phase
space from this physical perspective.
The odd-dimensional analog of symplectic geometry, called contact geometry,
comes up when we consider real-valued functions dened on the phase space. It
gives us the ability to solve rst-order PDEs in a manner that is both eective,
and consistent with these constraints.
1
1. Symplectic and Contact Geometry
1.1 Symplectic Manifolds
Before we attempt to understand the more general idea of a symplectic structure
on a manifold, we'll rst dene this notion for vector spaces. Suppose we begin
with an arbitrary nite-dimensional real vector space V along with an exterior
2-form ! on it, that is, a skew-symmetric bilinear map
! :VV !R:
1.1 Denition. We say that the form! is nondegenerate if there is no nonzero
vector u2V such that
!(u;v) = 0 for all v2V:
In this case, (V;!) is called a symplectic space, and ! its symplectic form.
As an important example, consider the even-dimensional spaceR
2n
, with the
suggestively-labeled coordinates (x;p) = (x
1
;:::;x
n
;p
1
;:::;p
n
). The canonical
symplectic form
!
0
:= dx^ dp =
n
X
j=1
dx
j
^ dp
j
indeed givesR
2n
the structure of a symplectic space. In fact, for any symplectic
space, we can nd an appropriate basis with respect to which the symplectic
structure takes this canonical form. This is the content of the following propo-
sition.
1.2 Proposition. If (V;!) is a symplectic space, then there's an isomorphism
of vector spaces : R
2n
! V that identies ! with the canonical symplectic
form !
0
, i.e.
! =!
0
:
Recall that
! is dened as
!(;) :=!( (); ());
and is called the pullback of! along . Hence we have the following commutative
diagram.
R
2n
R
2n
VV
R
!=!0
!
Corollary. As an immediate consequence, we see that every symplectic space
is necessarily of even dimension.
2
1. Symplectic and Contact Geometry
It follows from the proposition thatV admits a basisfx
1
;:::;x
n
;p
1
;:::;p
n
g,
called a Darboux basis, such that
!(x
i
;x
j
) = 0; !(p
i
;p
j
) = 0; !(x
i
;p
j
) =
ij
for alli;j. Then the setfdx
1
;:::; dx
n
; dp
1
;:::; dp
n
g constitutes a dual basis for
V
. As a result, we may replace the nondegeneracy assumption in our denition
of a symplectic space with the following equivalent condition.
1.3 Proposition. A 2-form ! on a vector space V is a symplectic form if and
only if the 2n-form !
^n
is nonvanishing, that is
!
^n
:=!^:::^!6= 0:
Proof. Suppose (V;!) is a symplectic space. Then by Proposition 1.2, we may
write ! = dx^ dp upon appropriate choice of basis, and hence
!
^n
=n! dx
1
^ dp
1
^:::^ dx
n
^ dp
n
6= 0:
On the other hand, suppose! is degenerate and letu
1
2V be a nonzero vector
with !(u
1
;v) = 0 for all v 2 V . We may complete this vector to a basis
fu
1
;:::;u
2n
g for V , and then !
^n
(u
1
;:::;u
2n
) = 0. But the volume form on
V is nonzero on any basis and unique up to multiplication by a nonzero scalar
since
dim
2n
^
V
=
2n
2n
= 1:
So we must actually have that !
^n
= 0.
Now, just as we have a notion of orthogonality within a vector space equipped
with an inner product, we may similarly dene skew-orthogonality for vectors
and subspaces within a symplectic space (V;!).
1.4 Denition. Two vectors u and v in V are said to be skew-orthogonal if
!(u;v) = 0:
The skew-orthogonal complement of a k-dimensional subspace UV , denoted
by U
!
, is the (2nk)-dimensional subspace of V consisting of those vectors of
V that are skew-orthogonal to every vector of U. That is,
U
!
:=fv2V j!(u;v) = 0 for all u2Ug:
Unlike the usual orthogonal complement (say, dened by the scalar product
in Euclidean space), however, the skew-orthogonal complement of a subspace
may intersect that subspace non-trivially. Actually, a subspace can even be fully
contained within its skew-orthogonal complement.
1.5 Denition. A subspaceUV is said to be isotropic ifUU
!
, i.e. if all
of its elements are pairwise skew-orthogonal.
3
1. Symplectic and Contact Geometry
Note that! is highly degenerate onU; not only can we nd a nonzero vector
skew-orthogonal to every other vector, but in fact, every vector in U has this
property. (In particular, isotropic subspaces demonstrate that a subspace of a
symplectic space need not be symplectic itself.) For this reason, it's obvious that
U can't have dimension 2n, as this would mean that V is isotropic. However,
we can actually obtain a much better bound on the dimension of U.
1.6 Proposition. An isotropic subspace of V is at most n-dimensional.
Proof. Let U be a k-dimensional isotropic subspace. Then, UU
!
, and so
k = dimU dimU
!
= 2nk:
Therefore kn.
Remark. It can be shown thatn-dimensional isotropic subspaces do indeed exist.
We call such a subspace a Lagrangian subspace.
Having developed the theory of symplectic geometry for vector spaces, we'll
next extend it to manifolds in general by way of a dierential 2-form that locally
denes a symplectic structure. We begin with a manifold M equipped with a
dierential 2-form
!2
2
(M):
1.7 Denition. We say that ! is nondegenerate if at every point z2M, the
exterior 2-form
!
z
: T
z
M T
z
M!R
is nondegenerate. If ! is both closed and nondegenerate, then (M;!) is called
a symplectic manifold, and ! its symplectic form.
In particular, the tangent space at any point z of a symplectic manifold M
is itself a symplectic vector space when equipped with the 2-form !
z
. So, since
a manifold has the same dimensionality as any of its tangent spaces, it follows
that any symplectic manifold is also even-dimensional.
In a similar manner, we'll see that many of the other results we'd established
for symplectic spaces will also extend naturally to manifolds. For instance, as
a consequence of Proposition 1.3, we have the following.
1.8 Proposition. A closed dierential 2-form! on a 2n-manifoldM is a sym-
plectic form if and only if the 2n-form !
^n
vanishes nowhere on M, that is,
if
!
^n
:=!^:::^!6= 0:
Remark. This result allows us to equivalently dene a symplectic manifold by
requiring!
^n
to be a volume form. We'll take a similar approach when we later
dene contact manifolds.
4
1. Symplectic and Contact Geometry
1.9 Denition. A submanifoldUM is called isotropic (resp. Lagrangian) if
at every point z2U, the tangent space T
z
U is an isotropic (resp. Lagrangian)
subspace of T
z
M.
It's a direct consequence of Proposition 1.6 that an isotropic submanifold of
M has dimension at mostn, and those of dimensionn are exactly the Lagrangian
submanifolds. We'll come across some examples of these kinds of submanifolds
in later sections.
Now, recall that we'd seen earlier that any symplectic vector space had the
same symplectic structure as (R
2n
;!
0
). Before we make an analogous statement
about symplectic manifolds, we'll dene more generally what we mean when we
say that two manifolds have the same symplectic structure.
1.10 Denition. Let (M;!
M
) and (N;!
N
) be symplectic manifolds, and a
dieomorphism M! N. We dene the pullback of !
N
along as the 2-form
given at any point z2M by
(
!
N
)
z
(;) := (!
N
)
(z)
(d
z
(); d
z
()):
Then, we call a symplectomorphism if
!
N
=!
M
;
and we say that (M;!
M
) and (N;!
N
) are symplectomorphic. In this case, we
thus obtain the following commutative diagram for every z2M.
T
z
M T
z
M T
(z)
N T
(z)
N
R
(
!
N
)z =(!
M
)z
d zd z
(!
N
)
(z)
This gives us the language necessary to generalize Proposition 1.2, the so-
called \linear version" of Darboux's theorem, and extend it to the case of sym-
plectic manifolds.
1.11 Theorem (Darboux). Let (M;!) be a symplectic manifold. Then for any
point z of M, we can nd neighborhoods UM of z and U
0
R
2n
of 0 such
that (U;!) and (U
0
;!
0
) are symplectomorphic.
An equivalent formulation of this theorem is that, about each point z2M,
there's a system of local coordinates (x;p) = (x
1
;:::;x
n
;p
1
;:::;p
n
), which we
may sometimes call Darboux coordinates, with respect to which
! = dx^ dp:
We've now encountered our rst example of a symplectic manifold, namely
(R
2n
;!
0
). We should observe thatR
2n
is naturally isomorphic to the cotangent
5
1. Symplectic and Contact Geometry
bundle ofR
n
. More generally, we nd that we may place a symplectic structure
on the cotangent bundle (which we thought of as the phase space) of an arbitrary
manifold as well.
Let's return to the n-manifold B from the beginning of this chapter. We'd
locally described its phase space T
B by the 2n-dimensional coordinate system
(x;p). By what we just observed, we may equip T
B with the dierential 2-form
given locally by ! = dx^ dp, thereby giving it the structure of a symplectic
manifold.
We'll see in the coming sections that this symplectic structure is actually
induced in a natural way from certain contact structures that are intrinsically
found in higher- or lower-dimensional spaces related to the phase space of our
manifold.
6
1. Symplectic and Contact Geometry
1.2 Contact Manifolds
Now that we've explored the theory of symplectic geometry and its relationship
to the phase space, we'll shift our attention to contact geometry. This time, we
begin with a (2n + 1)-manifold M equipped with a dierential 1-form
2
1
(M):
1.12 Denition. If the (2n + 1)-form ^ (d)
^n
is nowhere-vanishing on M,
that is, if it satises the nondegeneracy condition
^ (d)
^n
=^ d^:::^ d6= 0;
then we call a contact form.
Remark. Note the obvious similarity of this denition to Proposition 1.8, which
we observed was an equivalent denition of a symplectic structure.
As a 1-form, denes at each point z2M an exterior 1-form, i.e. a linear
map,
z
: T
z
M!R. Thus we can nd a vector v2 T
z
M such that
z
() =h;vi
z
:
The kernel of
z
is a hyperplane, called the contact plane at z, in the tangent
space T
z
M. We thereby obtain a eld of hyperplanes distributed over M,
given at each point z2M by
z
:= ker
z
T
z
M:
1.13 Denition. In this case, we call (M;) a contact manifold, and the hy-
perplane eld is called its contact structure.
Remark. The nondegeneracy condition placed on guarantees that this eld is
maximally nonintegrable, meaning that it has no integral submanifolds whose
dimension is equal to that of the eld.
It's important to note that for any z 2 M, we may multiply
z
by any
nonzero real number while leaving its kernel (the contact plane) unchanged. As
such, two contact 1-forms and ~ dene the same contact structure if and only
if there is some function f :M!Rn 0 such that
=f ~ :
Recall that we considered two symplectic manifolds to be \the same" if there
was a dieomorphism between them that identied their symplectic forms with
each other. Because dierent 1-forms can dene the same contact manifold, it's
easier to instead base our denition of what it means for two contact manifolds
to be \the same" on their hyperplane elds.
7
1. Symplectic and Contact Geometry
1.14 Denition. Let (M;
M
) and (N;
N
) be contact manifolds, and a dif-
feomorphism M!N. Recall that we dene the pushforward of as the map
: TM! TN given at each point z2M by
(
)
z
() := d
z
():
Then, we call a contactomorphism if
M
=
N
;
and we say that (M;
M
) and (N;
N
) are contactomorphic. In this case, we
obtain the following commutative diagram.
TM TN
M
N
M N
Remark. Equivalently, for a choice of contact 1-forms
M
and
N
for our man-
ifolds, we could ask that there exist a function f :M!Rn 0 such that
N
=f
M
;
where
N
is the pullback of
N
along , dened analogously to the pullback
of a 2-form we'd seen earlier.
We now consider a key example of a contact manifold, namely the odd-
dimensional space R
2n+1
with coordinates (x;y;p) = (x
1
;:::;x
n
;y;p
1
;:::;p
n
).
The canonical 1-form
0
:= dypdx = dy
n
X
j=1
p
j
dx
j
denes a contact structure onR
2n+1
, which we denote by
0
. Just as Darboux's
Theorem asserted that all symplectic manifolds are locally equivalent to R
2n
with the canonical symplectic 2-form, it turns out that a similar result holds for
contact manifolds as well.
1.15 Theorem (Pfa). Let (M;) be a contact manifold. Then for any point
z2M, we can nd neighborhoods UM of z and U
0
R
2n+1
of 0 such that
(U;) and (U
0
;
0
) are contactomorphic.
Corollary. It follows immediately that any contact manifold is necessarily of
odd dimension.
8
1. Symplectic and Contact Geometry
Hence, about any point z2 M, there exists a system of local coordinates
(x;y;p) = (x
1
;:::;x
n
;y;p
1
;:::;p
n
) with respect to which = ker, where
takes the form of the canonical 1-form
= dypdx:
In this local system, the 2n-dimensional subspace with coordinates (x;p) may
be endowed with the canonical symplectic structure ! = dx^ dp by taking the
exterior derivative (together with a sign change) of ,
! =d = dx^ dp:
Earlier we'd seen this symplectic form when studying the phase space. There
exists a similar setting in which the canonical contact structure arises, namely
the bundle of 1-jets of a manifold, which may be thought of as a sort of 1-
dimensional extension of the cotangent bundle.
1.2.1 The 1-Jet Bundle
Let's once again go back to ourn-manifoldB. In solving ak-th order PDE of a
real-valued smooth function dened onB, it makes sense to reduce our space of
possible solutions to functions onB modulo derivatives of order strictly greater
than k.
Recall that two functions f and g on B are said to dene the same germ
at a point x2 B if there exists an open neighborhood U B of x such that
f
U
=g
U
. The space of germs of smooth functions at x is denoted by C
1
(x),
and stores \local information" about functions near x.
1.16 Denition. Given two germs f;g2 C
1
(x), we write f
k
x
g if for any
smooth curve
: (;)!B with
(0) =x, we have
(f
)
(r)
(0) = (g
)
(r)
(0)
for any integer 0rk. This denes an equivalence relation between germs
in C
1
(x), and we dene the quotient
J
k
x
(B;R) := C
1
(x)=
k
x
:
The equivalence class of a germ f in this quotient is called the k-jet of f at x,
and is denoted by j
k
x
f. We then dene
J
k
(B;R) :=
a
x2B
J
k
x
(B;R);
the k-jet bundle of B, which we'll often denote by J
k
B.
Upon choosing a system of local coordinates, to say that two functions dene
the samek-jet atx is precisely to say that theirk-th order Taylor approximations
at x agree. This is expressed in the following proposition.
9
1. Symplectic and Contact Geometry
1.17 Proposition. For two functions f;g 2 C
1
(x), we have j
k
x
f = j
k
x
g if
and only if for any choice of coordinate chart (U;) centered at x, given by
coordinates (x
1
;:::;x
n
), we have
@
r
(f
1
)
@x
i1
@x
ir
(0) =
@
r
(g
1
)
@x
i1
@x
ir
(0)
for any 1i
1
;:::;i
r
n and 0rk.
For this reason thek-jet of a function atx may be thought of as a coordinate-
independent version of a Taylor approximation. In particular, the 1-jet of
any function f, at least locally, becomes a polynomial in the 2n + 1 variables
(x;y;p) = (x
1
;:::;x
n
;y;p
1
;:::;p
n
), where y = f(x) and p
j
= (@f=@x
j
)(x).
Using this notation, we obtain a bijection J
1
B!R
2n+1
given by
j
1
x
f7! (x;y;p):
Notice that we'd earlier coordinatized the phase space T
B using (x;p), and
hence we now see that the 1-jet bundle is in some sense an extension of the
cotangent bundle,
J
1
B
= (T
B)R;
where the summand R corresponds to the function value y.
Remark. Although we'll be mainly concerned with the 1-jet bundle, it's worth
observing that a similar correspondence exists between the higher-order jet and
cotangent bundles of B as well.
J
0
B J
1
B J
k
B
BR (T
B)R (T
k
B)R
We nd that J
1
B is a natural setting in which to study (rst-order) PDEs
dened on B, which was our original goal. Such a dierential equation is an
expression relating a functionf ofx to its rst-order partial derivatives; as such,
it can be written in the form
(x;y;p) = 0
for some smooth function : J
1
B! R. From this perspective, however, we
notice that the PDE may simply be viewed as a level set of this function, in
this example the preimage of zero under . In light of this observation, we may
oer an alternate denition of a PDE.
1.18 Denition. A rst-order PDE on B is a hypersurface V J
1
B.
10
1. Symplectic and Contact Geometry
Studying the geometry of this hypersurface will in fact allow us to nd
solutions of the PDE. To do this, we'll often consider a natural subset of J
1
B
that's associated with every function f dened on B.
1.19 Denition. Let f be a real-valued function dened on B. Then, the
1-graph of f is the n-dimensional subset
1
f
:=f(x;y;p)jx2Bg J
1
B:
Remark. The 1-graph is an extension of the usual set-theoretic graph of f that
also includes the values of the function's partial derivatives at each point. A
function solves a PDE if and only if its 1-graph is a submanifold of the corre-
sponding hypersurface V ; this is more or less obvious, since the 1-graph will be
contained in the PDE exactly when the associated function satises the con-
straints imposed by V .
Remarkably, the denition of a 1-graph imposes a sort of \velocity restric-
tion" on the 1-jet bundle that, in a natural way, gives rise to the canonical
contact structure we'd seen before. This happens as follows.
We begin with an arbitrary point z = (x;y;p)2 J
1
B. This point belongs to
the 1-graph of any function that, at positionx, has a value ofy and is changing
instantaneously with momentum p. In order to locally remain on such a 1-
graph, any instantaneous \velocity" (dx; dy; dp)2 T
z
J
1
B must be tangent to
the 1-graph. By denition the 1-graph, since p = dy=dx, this velocity must
satisfy the condition dy = pdx. In other words, it must lie in the kernel
z
of
the canonical 1-form
= dypdx
at the point z.
Figure 1.2: the contact plane at z
11
1. Symplectic and Contact Geometry
1.20 Denition. The eld TJ
1
B of contact planes corresponding to the
canonical 1-form is called the canonical contact structure or Cartan distribu-
tion on J
1
B.
The space (J
1
B;) hence becomes a contact manifold. From the gure, it's
evident that every contact plane may be coordinatized by (x;p); we simply
project it onto the (x;p)-hyperplane in J
1
B, which amounts to \forgetting" the
y-component of J
1
B,
J
1
B
= (T
B)R T
B:
Thus we obtain at each point z2 J
1
B an equivalence
z
= T
z
B:
Remark. We'd encountered earlier the fact that the canonical 1-form induces
a symplectic structure on the cotangent bundle T
B. In this context, this
symplectic form
! =d = dx^ dp
renders each 2n-dimensional contact plane (
z
;!
z
) a symplectic space. As a
result, (T
B;!) becomes a symplectic manifold (as expected), whose dimension
is 1 less than that of the contact manifold from which it was derived. For this
reason, we sometimes refer to the 1-jet bundle J
1
B as the contactization of the
phase space T
B.
(T
B)R
T
B J
1
B
B
contactization
Now, it's obvious by the way we constructed the canonical contact structure
that the tangent to any 1-graph passing through z is completely contained in
the contact plane
z
. In fact,
z
is, almost by its very denition, the closure of
the union of all such tangents. It follows that, given any function f on B, its
1-graph
1
f
is an n-dimensional integral manifold for ; it turns out that this
is the largest possible dimension for an integral manifold of a 2n-dimensional
contact structure.
1.21 Denition. In general, a maximal-dimensional integral manifold of a eld
of contact planes is called a Legendre manifold.
12
1. Symplectic and Contact Geometry
Any path through an integral manifold of is guaranteed to respect the
velocity condition on J
1
B described earlier, since its tangents are subspaces of
the contact structure.
Figure 1.3: the contact plane preserves the velocity condition on J
1
B
As we noted earlier, the 1-graph of any solutionf to a PDE will be a subman-
ifold of its associated hypersurface, and will additionally be an integral manifold
of the canonical contact structure. Hence, when attempting to construct such
solutions, we'll look for 1-graphs that conform to the restrictions imposed by
both of these structures.
1.2.2 The Cosphere Bundle
We'll now turn our attention to another naturally-arising contact structure,
which we'll make use of later on. Suppose we replace our manifold B with the
(n + 1)-dimensional \space-time" manifold M := BR, coordinatized locally
by z = (z
0
;:::;z
n
), with z
0
=t representing time and (z
1
;:::;z
n
) representing
the physical space B.
13
1. Symplectic and Contact Geometry
At each point z2 M, the cotangent space T
z
M consists of instantaneous
space-time \trajectories" q = (q
0
;:::;q
n
), which are composed of a time com-
ponentq
0
and the usual physical momentum component (q
1
;:::;q
n
). Then the
phase space ofM is the (2n + 2)-dimensional cotangent bundle T
M with local
coordinates (z;q).
Indeed, we may visualize the movement of a particle traveling throughB over
time as a geometric curve within M. Then at any point on the curve, the time
componentq
0
of the particle's space-time trajectory corresponds to its speed at
that point. In some sense, this information is irrelevant, since it's dependent on
the scaling of our time axis R. In other words, we're only concerned with the
direction ofq, and not with its magnitude. Hence, we may simplify our analysis
of this space by projecting the set of all velocities in a given direction to a single
direction vector.
1.22 Denition. LetV be an arbitrary vector space. We writevw ifv =w
for some 2Rn 0, and thereby dene the projectivization of V as
PV := (Vn 0)=:
The spherization of V , denoted by SV , is dened similarly, this time letting
vw if v =w for some 2R
+
n 0.
Remark. We may think ofSV as having been obtained fromPV by additionally
equipping each vector with a co-orientation (essentially a positive or negative
sign).
Remark. Provided that we have a norm on V , we may elect the unit norm
vectors as representatives for their respective equivalence classes in PV . As
such, PV takes on the appearance of a half-sphere in V . Likewise, we may
identify SV with the full sphere. From this, it's easy to see that
dimPV = dimV 1;
and similarly for SV .
Now, let z be a point of M. We call a hyperplane (passing through 0) in
the tangent space T
z
M a contact element at z. As before, we may think of a
normal vectorq2 T
z
M to this plane as an element of the cotangent space T
z
M
by viewing it as the linear functional
^ q() :=h;qi
z
:
By doing this, we may represent the 1-dimensional family of normal vectors
to this plane by a single equivalence class of covectors in PT
z
M, and hence
obtain a one-to-one correspondence between the contact elements at z, and
the projectivized cotangent space PT
z
M. For this reason, we may refer to an
element (z;q)2PT
z
M as either a contact element or a covector, depending on
the context.
14
1. Symplectic and Contact Geometry
1.23 Denition. We now dene the cosphere bundle or manifold of contact
elements of M as the bundle
PT
M :=
a
z2M
PT
z
M:
The sphere bundle or manifold of co-oriented contact elements of M, denoted
by ST
M, is analogously dened.
Remark. For simplicity, we'll mainly focus on the cosphere bundle ofM; almost
every result and construction in this setting has a virtually identical counterpart
in the sphere bundle.
Just as we did with the cotangent bundle T
M, we may locally coordinatize
the cosphere bundle by (z;q). However, we now regard the q-component as
being described by a system of homogeneous coordinates q = [q
0
: : q
n
],
where q q for any 2 Rn 0. As a result, we see that PT
M is (2n + 1)-
dimensional.
Figure 1.4: the projectivization PT
z
M
We'll now nd that there's a natural skate condition that restricts the di-
rections in which contact elements in PT
M may move; it's analogous to the
velocity condition in the 1-jet bundle J
1
B. The idea is that any contact element
(z;q) behaves like a \skate" moving over a rink in that it may rotate in place
and slide along its own direction, but may not \skid" in the direction normal
to itself.
Figure 1.5: the skate condition
15
1. Symplectic and Contact Geometry
More specically, the skate condition asserts that if we wish to travel in the
tangent space of PT
M at (z;q), say with some trajectory (dz; dq), then the
projection dz2 T
z
M must remain normal to the vector determined by q, i.e.
we must have
qdz = 0:
In dq-space, we may move in any of the n available directions since the
projection discards the dq component anyway. In the (n + 1)-dimensional dz-
space, we have 1 less degree of freedom, since we may not move in the direction of
the normal vector. Hence, we obtain a 2n-dimensional subspace of T
(z;q)
PT
M
of allowed directions of movement. This subspace may be equivalently viewed
as the kernel
(z;q)
of the tautological 1-form
:=qdz
at the point (z;q).
1.24 Denition. The eld TPT
M of contact planes corresponding to the
tautological 1-form is called the tautological contact structure on PT
M.
This turns the cosphere bundle (PT
M;) into a contact manifold. Further-
more, we can use the tautological 1-form to dene a symplectic structure, in
a very similar manner as we'd done in the previous section with the canonical
1-form on J
1
B. To do this, we simply \unprojectivize" PT
M via the natural
inclusion map sending a projectivized vector [q
0
: :q
n
] to the representative
(q
0
;:::;q
n
) of its equivalence class having unit norm,
PT
M ,! T
Mn 0
M
:
Remark. Here, we the remove the zero section 0
M
T
M, which consists of
all elements of the form (z; 0), since no projectivized cotangent space PT
z
M
contains the zero vector.
The required symplectic form, which looks like the canonical symplectic
form, is one again given by the exterior derivative of ,
! :=d = dz^ dq:
We thus turn the phase space (T
Mn 0
M
;!) into a symplectic manifold. In
this case, however, we nd that the dimension of the symplectic manifold is 1
higher than that of the contact manifold, not 1 lower as it was before. So, we
call T
Mn 0
M
the symplectization of PT
M.
PT
M T
Mn 0
M
M
symplectization
16
1. Symplectic and Contact Geometry
In the context of the 1-jet bundle of B, we observed that any solution of
a PDE necessarily conformed, in some way, to the natural velocity restriction
in J
1
B; namely, its 1-graph was an integral manifold of the canonical contact
structure.
Likewise, any path through an integral manifold of the tautological contact
structure is guaranteed to obey the skate condition. Paths that aren't integral
to this eld will at some point involve a skidding contact elements, and so we're
not interested in them.
Figure 1.6: the tautological plane preserves the skate condition in PT
M
We conclude this section by taking a look at some examples of integral
manifolds of the tautological contact structure, which arise from submanifolds
within M itself.
1.25 Example. Let HM be a hypersurface. At any point z2H, consider
the covector (z;q) that's normal to the tangent space T
z
H. Then the collection
over H of all covectors constructed in this way is an integral manifold of ,
since at any of its points (z;q), any trajectory (dz; dq) through this manifold
has dz2 T
z
H, which is by denition normal to q.
17
1. Symplectic and Contact Geometry
Figure 1.7: the integral manifold in Example 1.25
1.26 Example. Now, let z be a point of M, and consider the collection of all
contact elements at z. Sincez is held constant, any trajectory (dz; dq) through
this manifold has dz = 0, which is orthogonal to anyq. So, this manifold is also
integral to .
Figure 1.8: the integral manifold in Example 1.26
Note that in Example 1.25, we had all n degrees of freedom in choosing z
within the hypersurface H, but we had no freedom in choosing q since it was
completely determined by the geometry of H. On the other hand, in Example
1.26, z was xed while we had all n degrees of freedom in choosing q. So in
both cases, the integral manifold obtained is n-dimensional. Now consider the
following example.
1.27 Example. Let H be a k-dimensional submanifold of M. The collection
over H of all contact elements tangent to H is an integral manifold of .
Remark. Setting k =n gives us the rst example, and k = 0 the second.
In this more general case, we have k degrees of freedom when choosing z
within H, and nk degrees of freedom in choosing a normal vector q for the
k-dimensional tangent space T
z
H. Hence, we again obtain an n-dimensional
integral manifold; in fact, it's a Legendre manifold, since the tautological contact
structure is 2n-dimensional.
18
1. Symplectic and Contact Geometry
1.2.3 Relating the Contact Structures
By now, we've noticed that the constructions of the two \natural" contact struc-
tures share many similarities. In this section, we'll formally relate the canonical
and tautological contact structures, and see how they're actually, in some re-
gard, the same.
Remark. For this section, we'll consider the tautological contact structure de-
ned instead on the sphere bundle; it's more or less identical to the one on the
cosphere bundle.
1.28 Theorem. There exists a contactomorphism
ST
R
n
! J
1
S
n1
that identies the tautological contact structure in ST
R
n
with the canonical
contact structure in J
1
S
n1
.
Remark. As a consequence we obtain the following diagram, which summarizes
the relationships between the spaces involved.
J
1
S
n1
ST
R
n
T
S
n1
T
R
n
n 0
R
n
S
n1
R
n
contactization
symplectization
Proof. Consider a point (z;q)2 ST
R
n
, which represents a contact element
placed at z2 R
n
and oriented by a vector q2 S
n1
. We associate with this
contact element a point (x;y;p)2 J
1
S
n1
via the map
x =q; y =zq; p =z (zq)q:
This function is easily veried to be a dieomorphism, its inverse being given
by the map
z =p +yx; q =x:
Then we have
dypdx = (qdz +zdq) (z (zq)q)dq
=qdz +zdqzdq + (zq)qdq
=qdz;
so that the canonical contact structure given by dy = pdx is equivalent to the
tautological contact structure given by qdz = 0.
19
1. Symplectic and Contact Geometry
Geometrically, the correspondence in the theorem translates to the following.
We use the positionx on the sphere to determine the orientationq for the contact
element placed at z. The normal component of z is obtained by rescaling the
normal vectorx by the function valuey, and the tangential component is given
by the gradient vector p2 T
x
S
n1
.
Figure 1.9: the contactomorphism in Theorem 1.28
In the proof of the theorem, we veried algebraically that this correspondence
preserves the contact structure. However, this fact can also be seen geometri-
cally from the diagram above, in the following way.
Assume that we wish to move a contact element (z;q) locally via some
arbitrary trajectory (dz; dq). We can break this movement into two steps: a
movement in which q is xed, followed by one in which z is xed.
Suppose rst that we x q, and move z to some new position z
0
= z + dz.
Then we have dq = 0, and the velocity condition dy =pdx on the 1-jet bundle,
which denes the canonical contact structure, becomes
dy =pdx =pdq = 0:
This implies that y is to remain constant. From the diagram, we can see that
changing the function valuey corresponds to changingz in the normal direction,
that is, the direction ofq. So ify isn't supposed to change, thenz can't move in
the normal direction; in other words, dz can't have a component in the direction
of q, and we necessarily have
qdz = 0:
This is precisely the skate condition that denes the tautological contact struc-
ture on the sphere bundle ST
R
n
.
20
1. Symplectic and Contact Geometry
Figure 1.10: a movement of (z;q), with q remaining xed
Suppose next that we x z, and change q to some new value q
0
= q + dq.
Then the change in the value of y = zq is given by dy = zdq. But, the
only component of z that contributes to the value of this scalar product is the
tangential component, namely p. In other words, we have
dy =zdq =pdq =pdx;
which is exactly the velocity condition we started with. Hence we can move q
in any direction dq at all, while still preserving the velocity condition; this once
again agrees with the skate condition.
Figure 1.11: a movement of (z;q), with z remaining xed
21
Chapter 2
Applications to First-Order PDEs
Recall that, given an n-manifold B representing physical space, a rst-order
PDE of functions dened on B can be viewed as a hypersurface V in the 1-
jet bundle J
1
B. Now that we've explored the canonical contact structure for
J
1
B, we can make use of it to locally construct integral manifolds of V , which
correspond to 1-graphs of solutions of the equation itself. This gives us a way
of solving PDEs geometrically.
To this end, we begin with a point z2V . Consider the intersection of the
tangent space T
z
V to the PDE at z and the contact plane
z
at z. Both of
these are hyperplanes (passing through 0) in T
z
J
1
B. As such, this intersection
is either 2n-dimensional, if these hyperplanes are equal, or (2n 1)-dimensional
if they aren't.
2.1 Denition. The point z is said to be a singular point of V if we have
T
z
V =
z
. Otherwise, it's said to be a regular point of V .
Remark. In general, singular points represent some sort of a degenerate condi-
tion, and so we're not interested in them. Moreover, the singular points of V
are isolated.
Figure 2.1: a regular point z2V
Any instantaneous trajectory (dx; dy; dp) within the intersection T
z
V\
z
will simultaneously respect the velocity conditions imposed by the specic PDE,
by remaining tangent to V , and the velocity condition intrinsic to the ambient
space J
1
B, by remaining within the canonical contact structure.
As we noted before, these are exactly the properties that characterize any
solution of the PDE. So to construct a solution, we'll attempt to stitch together
1-dimensional \bers" whose tangents are such trajectories. This process is
described in the following section.
22
2. Applications to First-Order PDEs
2.1 The Characteristic Field
Supposez2V is a regular point. Then, the intersection T
z
V\
z
is a (2n 1)-
dimensional hyperplane in the 2n-dimensional contact plane
z
, which we'd
earlier equipped with the canonical symplectic form
! = dx^ dp:
Using!, we're able to determine a 1-dimensional direction within this intersec-
tion, as was our goal.
2.2 Proposition. The skew-orthogonal complement of a line U (that passes
through zero) in a symplectic space is a hyperplane containing this line.
Proof. Suppose U is spanned by u. The skew-orthogonal complement of U is
the hyperplane U
!
=fvj !(u;v) = 0g. Since !(u;u) =!(u;u), we have
!(u;u) = 0, and so u2U
!
.
As a result, we see that T
z
V\
z
contains a distinguished line, namely its
skew-orthogonal complement (T
z
V\
z
)
!
.
2.3 Denition. The direction of this line is called the characteristic direction
at z. We thus obtain a eld of characteristic directions dened at the regu-
lar points of V , called the characteristic eld, whose integral curves are called
characteristics.
Figure 2.2: the characteristic direction at z
At each point, we may also refer to a vector spanning the line in the charac-
teristic direction as a characteristic vector. We can actually obtain a closed-form
expression of this vector if given the PDE that denes V . As before, suppose
that this PDE is
(x;y;p) = 0;
where is some smooth function J
1
B!R.
23
2. Applications to First-Order PDEs
2.4 Theorem. The characteristic vector (x
0
;y
0
;p
0
) at a regular point z2V is
given in components by
x
0
=
@
@p
; y
0
=p
@
@p
; p
0
=
@
@x
@
@y
p:
Remark. In fact, this formula denes a vector eld on all of J
1
B, not just the
hypersurface V .
Proof. In what follows, we'll use (x;y;p) as coordinates for the tangent space
T
z
J
1
B. The desired characteristic vector belongs to the tangent space T
z
V to
the hypersurface at that point, and hence it must satisfy the equation obtained
by dierentiating the PDE,
@
@x
_ x +
@
@y
_ y +
@
@p
_ p = 0: (2.1)
Moreover, the characteristic vector belongs to the contact plane
z
at z, and
thus also satises the equation that denes the canonical contact structure on
the 1-jet bundle,
_ y =p _ x: (2.2)
To obtain an expression that denes the intersection T
z
V\
z
, we combine the
equations (2.1) and (2.2),
@
@x
+
@
@y
p
_ x +
@
@p
_ p = 0: (2.3)
By denition, the characteristic eld was obtained as the skew-orthogonal
complement, within the contact plane
z
, of this intersection with respect to !.
In previous sections, we saw that (x;p) constitutes a coordinate system for
z
upon applying a projection that \forgets" the y-component of T
z
J
1
B. To aid
in the calculation of this complement, we note that given two arbitrary vectors
(x
1
;p
1
); (x
2
;p
2
)2
z
, we have
!((x
1
;p
1
); (x
2
;p
2
)) = dx^ dp((x
1
;p
1
); (x
2
;p
2
))
=x
1
p
2
x
2
p
1
;
where x
1
p
2
and x
2
p
1
denote scalar products. If these two given vectors are in
particular skew-orthogonal, then we have
x
1
p
2
x
2
p
1
= 0:
This is exactly the form of equation (2.3), from which we can identify thex
0
-
andp
0
-components of the characteristic vector spanning the line (T
z
V\
z
)
!
as
the coecients of _ p and _ x, respectively. Finally, we can refer to equation (2.2)
to obtain the y
0
-component as well.
24
2. Applications to First-Order PDEs
2.1.1 Construction of Solutions
The characteristics corresponding to a PDE V are precisely the desired 1-
dimensional \bers" we mentioned earlier; they can be put together to locally
form the 1-graphs of solutions, as expressed by the following theorem.
2.5 Theorem. If a characteristic ofV passes through a pointz of a 1-graph
of a solution of the PDE given by V , then it's entirely contained in .
Proof. Firstly, note that if is the 1-graph of such a solution, then we have
T
z
T
z
V . Moreover, T
z
z
, since any 1-graph is an integral manifold (in
fact, a Legendre manifold) of the canonical contact structure . Hence T
z
is
contained in the intersection T
z
V\
z
. So if u is a vector in the characteristic
direction at z, then by denition of the characteristic eld, it must be skew-
orthogonal to T
z
V\
z
, and consequently, to T
z
as well.
Now let be the canonical 1-form that denes. Then since is a Legendre
manifold of , we have that
= 0. Recall that ! was obtained by taking the
exterior derivative of . Thus
!
=d
= 0
and so T
z
z
is an isotropic subspace. So if u didn't lie in T
z
, then
the span of u and T
z
would be an (n + 1)-dimensional isotropic subspace of
z
, contradicting Proposition 1.6. Therefore the characteristic vector at each
point of V must be tangent to , and it follows that the entire characteristic is
contained in .
Corollary. As a consequence of this theorem, we see that the 1-graph of any
solution of this PDE can be bered into characteristics.
We'll now turn our attention to how we may obtain such a 1-graph corre-
sponding to a solution of a PDE given certain initial conditions. Suppose we
begin with an (n1)-dimensional submanifold
B, and a smooth real-valued
initial function
f
0
:
!R:
2.6 Proposition. The image of
underf
0
is an (n 1)-dimensional isotropic
submanifold of the hypersurface V .
Let's denote this submanifold by ~
, and the characteristic eld associated to
our PDE by v. To avoid points of degeneracy, we'll assume that at any point
z2 ~
, the characteristic throughz is not tangent to ~
(that is, the characteristic
vectorv
z
atz is not contained in the tangent space T
z
~
). Then, the span ofv
z
and T
z
~
is ann-dimensional subspace of T
z
V . This is pictured in the following
diagram.
25
2. Applications to First-Order PDEs
Figure 2.3: the n-dimensional space spanned by vz and Tz ~
2.7 Theorem. The union of the characteristics at each point of ~
is locally the
1-graph of a solution f of the PDE corresponding to V .
Proof. Let V be the n-dimensional submanifold which is the union of all
characteristics passing through ~
. The Lie derivative of the contact 1-form in
the direction of this eld is given by
L
v
=
v
d + d(
v
);
where we have
v
=(v) and
v
d() = d(v;). Letz be a point on ~
. Firstly,
v
z
lies in the contact plane dened by
z
= ker
z
, and hence in general
v
=(v) = 0: (2.4)
Next, for any u2 T
z
V , we have d
z
(v
z
;u) =!
z
(v
z
;u) = 0 by denition of
the characteristic eld. So in general
v
d() = d(v;) = 0: (2.5)
When combined, the equations (2.4) and (2.5) imply that L
v
= 0; in other
words, the canonical 1-form is constant along the characteristic eld v, and
the same holds for ! =d.
Now for any pointz2 ~
, the tangent space T
z
T
z
V is isotropic since it's
spanned by v
z
, which is the characteristic vector, and T
z
~
, which is isotropic.
Therefore since ! is constant along v, then in fact T
z
0 is isotropic for any
arbitrary point z
0
2 . As a result,
= 0, and hence V is a Legen-
dre manifold, i.e. an n-dimensional integral manifold of the canonical contact
structure . If is given locally in coordinates by (x;y;p), where y =f(x) and
p
j
=g
j
(x) for some functions f and g
j
, then
= (dypdx)
=
n
X
j=1
@f
@x
j
dx
j
n
X
j=1
g
j
dx
j
= 0;
whereby we obtaing
j
=@f=@x
j
. Thus is actually 1-graph, namely the 1-graph
of the function f.
26
2. Applications to First-Order PDEs
To summarize, we're able to construct a solution of a PDE of functions on
B in the following way. We begin by considering the PDE as a hypersurface
V in the 1-jet bundle J
1
B. To this hypersurface we associate its characteristic
eld.
Given an initial function f
0
on a submanifold
B, we consider the sub-
manifold V created from the 1-dimensional characteristics passing through
the lift ~
V of
under f
0
. As a consequence of the theorem, is in fact the
1-graph of some function f dened on B. And since is a submanifold of V ,
then f is actually a solution of the desired PDE.
Figure 2.4: the construction of a solution locally
27
2.2 Wave Propagation
We can now apply the techniques we've developed thus far to study problems in
the propagation of waves over manifolds. Let's return to the (n+1)-dimensional
space-time manifoldM =BR. Recall that we'd equipped its cosphere bundle
PT
M with the tautological contact structure. In previous sections, we'd seen
that a hypersurface in the contact manifold J
1
B represented a rst-order PDE.
We'll now make a similar observation about the cosphere bundle.
We begin by considering the set of all unit-speed curves passing through a
point x2 B. In M, the tangent to any one of these curves g at some point
z = (x;t) is a line whose \physical" displacement dg and \time" displacement
dt are equal in magnitude. Varying through all of these curves, these tangents
form a \cone of possible directions" in the tangent space of z.
2.8 Denition. The cone atz given bykdgk
x
=jdtj is called the light-cone at
z, and is denoted by C
z
. We denote this eld of cones over M by C.
Remark. The top half of a light-cone is sometimes called the future cone, as
it represents the set of all points of space-time that could be reached from the
vertex by a unit-speed \light-beam." Similarly, the bottom half is sometimes
called the past cone.
Figure 2.5: the light-cone at z
The eld of light-cones describes the motion of light particles in a vacuum-
like medium; they all travel with the same speed regardless of position, time,
or direction. In an arbitrary medium, however, the speed of a moving particle
may depend both on where the particle is (in space-time) and on the direction
of propagation.
28
2. Applications to First-Order PDEs
To model this, we may instead assign to each such directionq atz = (x;t) a
dierent velocity v(q). Then the tangent to the motion in this direction slants
upward or downward. If v changes smoothly with the parameter q, then we
obtain a distorted \wae-cone" at z given by
kdgk
x
=vjdtj:
This cone is denoted by C
z
(v), and this eld of cones over M by C(v).
Remark. Note that this direction atz may be equivalently viewed as a covector
that orients the contact element (z;q)2PT
M. As such, when dened over all
points z2M, the velocity v becomes a function
v :PT
M!R:
Now, at a point z2 M, we may consider the family of contact elements
(z;q)2 PT
M that are tangent to the cone C
z
(v) of possible velocities at z.
As we vary z through M, the collection of all such contact elements in PT
M
forms a hypersurface V PT
M in the cosphere bundle of M.
2.9 Denition. We call this hypersurface the Fresnel hypersurface.
Figure 2.6: the Fresnel hypersurface V
29
2. Applications to First-Order PDEs
Remark. Since we may think of a contact element (z;q) as a point of space-time
together with a direction in which to \perturb" that point, we may sometimes
also refer to contact elements in the cosphere bundle of the space-time manifold
M as retardations. For this reason, the Fresnel hypersurface may also be called
the retardation hypersurface corresponding to v.
In a medium that's given by a velocity function v : PT
M ! R, waves
propagate at each pointz2M along, or tangent to, the coneC
z
(v) of velocities
at that point. As such, we'll see that the wave resulting from a given initial wave
front may be viewed as an integral submanifold of the corresponding retardation
hypersurface V PT
M.
More specically, suppose we wish to consider the propagation of a wave
originating from an initial (n1)-dimensional wave front
in then-dimensional
physical space B. By virtue of being a hypersurface of B,
has a 1-parameter
family of tangent hyperplanes in T
z
M at each pointz, from which we may select
the unique hyperplane (z;q) that's also tangent to the cone of velocities at z.
This contact element (z;q), by its very denition, belongs to V .
Figure 2.7: choosing the appropriate contact element at z2
The collection taken over
of all such contact elements forms an integral
submanifold of V , as a consequence of Example 1.27. As time progresses, the
wave will propagate through the medium, giving rise to a large front, a hyper-
surface in M representing this propagation.
2.10 Denition. In the space-time manifold M, the isochrone corresponding
to a point in timet
0
2R is then-dimensional hypersurface which is the preimage
of t
0
under the canonical projection
M =BRR:
Remark. In the diagram above, the isochrones are given by horizontal planes.
We may think of the initial wave front
as the intersection of the isochrone
t = 0 and the large front arising from
. The instantaneous wave front at some
other time t
0
may be obtained as the intersection of the large front with the
corresponding isochrone.
30
2. Applications to First-Order PDEs
2.2.1 Huygens' Principle
We're now able to apply the theory of characteristics we'd developed in the
1-jet bundle to understand how exactly the large front is obtained from the
initial front
. Note rst that the contact elements that are tangent to the large
front all lie within the retardation hypersurface, by the assumption that they
represent the propagation of the wave front
through M.
As such, they form a Legendre manifold (that is, an n-dimensional integral
manifold of the tautological contact structure ) within the cosphere bundle
PT
M. Then as a consequence of Theorem 2.5, the characteristic through
any point of this integral manifold is itself completely contained in the integral
manifold. This is called Huygens' principle in wave propagation.
Thus, the large front can be constructed from
in the following way; the
technique is analogous to Theorem 2.7. We rst lift
to PT
M via the map
that associates to each point z2
the unique contact element tangent to both
and C
z
(v). We next consider the union of all characteristics passing through
these points (z;q)2PT
M and thereby obtain a Legendre manifold in PT
M.
The large front is then simply the image of this manifold under the canonical
projection PT
MM.
Figure 2.8: constructing the large front from the initial front
31
2. Applications to First-Order PDEs
An enlightening result of this construction is the following. Suppose that we
consider a pointz2
as an initial wave front in its own right, and associate to
it a Legendre manifold in PT
M (in this case, the lifting
associates multiple
contact elements to the pointz, since our initial front is no longer a hypersurface
in B). This manifold contains the characteristic passing through the lift of z
when we instead consider z as a point of
, as we'd done in the construction of
the large front.
This asserts that the propagation of waves originating atz, as a single point,
agrees in some way with those originating atz, as an element of
. In particular,
the instantaneous front at any time obtained from
is precisely the envelope
of the fronts at that time obtained from each of the points of
. Hence, we
obtain the same result whether we choose to think of
as a single wave front
which propagates \all at once," or as a collection of points which all propagate
individually.
32
References
[1] V. I. Arnold. Contact geometry and wave propagation. 1989.
[2] V. I. Arnold. Topological invariants of plane curves and caustics. University
Lecture Series, 1994.
[3] V. I. Arnold. Lectures on partial dierential equations. Universitext, pages
1{26, 2004.
[4] Ana Cannas da Silva. Lectures on symplectic geometry. 2006.
[5] Hansj org Geiges. Contact geometry. 2003.
[6] Ko Honda. Contact geometry. Lecture notes.
[7] Andrew D. Lewis. Holomorphic and real analytic jet bundles, 2014.
[8] Harry Smith. Symplectic and poisson manifolds.
33
Abstract (if available)
Abstract
In this paper, we'll develop a method of geometrically solving first-order partial differential equations (PDEs) of functions on manifolds. To do this, we'll begin by considering the phase space (cotangent bundle) of a manifold, and the natural, even-dimensional symplectic geometry associated with it. We'll then ""extend"" these ideas to odd-dimensional manifolds and consider the analogous theory of contact geometry that exists within them. In particular, we'll focus on two important contact manifolds that arise in the analysis of PDEs, namely the 1-jet bundle J¹B of a manifold B representing physical space, and the cosphere bundle ℙT*M of a manifold M = B⊕ℝ representing space-time. We'll conclude the paper by applying the theories we develop in these manifolds to actually construct solutions of PDEs, and take a look at examples in the theory of wave propagation.
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Sahakian, Alexander Andre
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Core Title
Applications of contact geometry to first-order PDEs on manifolds
School
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Master of Arts
Degree Program
Mathematics
Publication Date
04/06/2018
Defense Date
03/22/2018
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