Focal and parallel surfaces

For an embedding
**X**: *U* -> **R**^{3} of a domain
*U* of **R**^{2} into 3-space, the *parallel
surface* at distance *r* is the image of the *parallel
map*
**X**_{r}: *U* -> **R**^{3}
defined by
**X**_{r}(**P**) = **X**(**P**) + *r***N**(**P**),
where
**N** = (**X**_{x} x **X**_{y})/|**X**_{x} x **X**_{y}|
is the unit normal of **X**. The parallel surface can be thought
of as a "wave front" at distance *r* from the image of **X**.
Let *S*(**X**_{r}) be the set of singular
points of **X**_{r}.

S(**X**_{r}) = {**P** *U* | rank
*d*(**X**_{r}) < 2}

Let be the normal bundle of **X**:

= {(**V**, **P**) **R**^{3} x *U* | **V** is normal to **X** at **P**}

Gauss thought of his mapping **N** as assigning to each point of
a surface a point on the sphere at infinity, analogous to the
celestial sphere used in navigation and surveying [Ba, p. 45]. Let
*p*: **R**^{3} - {0} -> *S*^{2}
be radial projection. For *r* sufficiently large, the parallel
surface to **X** at distance r does not pass through the origin, so
*p* ° **X**_{r} is defined, and the limit
as *r* goes to infinity of (p ° **X**_{r})
is **N**. In other words, *the Gauss map is the parallel map at
infinity*. Another way to see this is to consider the family of
maps
*I*_{t}: *U* -> **R**^{3},
0 t < 1:

*I*_{t}(**P**) = (1-*t*) **X**(**P**) + *t* **N**(**P**)

= (1-*t*)[**X**(**P**) + *t*/(1-*t*) **N**(**P**)]

Now consider an immersion
**X**: *M*^{n} -> **R**^{n+1}
of the smooth *n*-manifold *M* in Euclidean
(*n* + 1)-space. Let be the unit
normal bundle of **X**:

= {(**V**, **P**) *S*^{2} x *M*|**V** is
normal to **X** at **P**}

**X**_{r}: -> **R**^{n+1},
**X**_{r}(**V**, **P**) = **X**(**P**) + *r***V**

For each point **A**
**R**^{n+1} let
*D*_{A}: *M* -> **R** be
the radial distance squared function from **A**:

*D*_{A} = |**A** - **X**(**P**)|^{2}

*D* : **R**^{n+1}n -> **R**^{n+1} x **R**,
*D*(**A**, **P**) =
(**A**, *D*_{A}(**P**))

To relate the map : -> **R**^{n+1} and the
Gauss map **N**: -> *S*^{n},
we fit the two families *D* and together,
following Looijenga [Lo], [Wa2, p.713]. Consider the family of
functions

*L*:
*S*^{n+1} x *M*^{n} -> *S*^{n+1} x **R**

*L*((*a*_{1}, ...,
*a*_{n+2}), **P**) =
*a*_{n+2}|**X**(**P**)|^{2} - 2
(*a*_{1}, ...,
*a*_{n+1}) ^{.} **X**(**P**)

=
*a*_{n+2}
*D*(1/(*a*_{n+2}(*a*_{1}, ...,
*a*_{n+1}), **P**) -
1/(*a*_{n+2}|(*a*_{1}, ...,
*a*_{n+1})|^{2}

**Theorem 6.1** (Looijenga). Let *M*^{2} be a
smooth surface. For an open dense subset *B* of the space of
immersions
**X**: *M*^{2} -> **R**^{3},
the germ at (**A**, **P**) of the family *L* is a
versal unfolding of the germ of *L*_{A} at
**P** for all (**A**, **P**) *S*^{3} x *M*^{2}.

**Proof** See [Lo] and [Wa2]. Looijenga proves the equivalent
statement that the germ of the mapping
*L*:*S*^{3} x *M*^{2} -> *S*^{3} x **R**
is stable. (the subset *B* consists of all immersions whose jet
extensions are transverse to a certain Whitney stratification of a jet
space).

For **X** *B*, the
singularities of the focal set **F** are cuspidal edges,
swallowtails, elliptic umbilics, and hyperbolic umbilics. *The
umbilic singularities occur precisely at the foci of the umbilic
points of the immersion*. A discussion of the geometry of the
focal set can be found in Porteous' paper [Por1], which was a starting
point for our research on the extrinsic geometry of surfaces.
Porteous calls the cuspidal edges of the focal surface *ribs*.
The corresponding curves on the surface M are the *ridges* of the
immersion **X** (cf. chapter 3). If the map : -> **R**^{3} has a cusp at
(**V**, **P**) , then **P** is a ridge point of **X**, and
1/|**V**| is the principal curvature associated with the ridge at
**P**.

This description does not include those ridge points with
associated principal curvature zero, which correspond to ribs at
infinity. To include these points, consider the bifurcation set
*F* S^{n+1} of
the family *L*. For the diffeomorphism
*F*: *S*^{n+1}_{+} -> **R**^{n+1},
*f*((*a*_{1}, ..., *a*_{n+1})) =
1/*a*_{n+2} (*a*_{1}, ...,
*a*_{n+1}), we have *f*(*F*^{~}
*S*^{3}_{+}) = *F*. For **X**
*B*, the singularities of
*F*^{~} are cuspidal edges, swallowtails, elliptic
umbilics, and hyperbolic umbilics. If the catastrophe map ^{~} of *L* has a cusp at
(**A**, **P**)
*S*^{n+1} x *M*^{n}, then
**P** is a ridge point of **X** with associated principal
curvature a_{n+2}/|(*a*_{1}, ...,
*a*_{n+1})|. The singularities of F^{~}
*S*^{n+1}_{0}
correspond to singularities at infinity of *F*.

For example, the focal surface of the monkey saddle has an elliptic
umbilic at infinity. (the monkey saddle is an example of an immersion
**X**: *M*^{2} -> **R**^{2}
such that **X** is in *B* but **X** is not in *A*.
The inclusion of the unit sphere in **R**^{3} is an example
of an immersion in *A* but not *B*.)

The following theorem implies theorem 3.1(d).

**Theorem 6.2** If **P** is a cusp of the Gauss map of
**X**: *M*^{2} -> **R**^{3},
then **P** is a ridge point of **X** with associated principal
curvature zero. If **X** *A*
then the cusps of the Gauss map of **X** are the only points of M
with this property.

**Proof** **P** is a cusp of the Gauss mapping of **X** if
and only if (**V**, **P**) is a cusp of the catastrophe map
of the family , where **V**
is a unit normal vector to **X** at **P**. The 2-parameter
family is equivalent to the restriction of the
3-parameter family *L* to
*S*^{3}_{0} x *M*^{2}.
therefore if has a cusp at
(**V**, **P**), then ^{~} has a
cusp at (*i*(**V**), **P**), where
i: *S*^{2} -> *S*^{3} is the
inclusion and is the catastrophe map of the
family *L*. So **P** is a ridge point of **X** with
associated principal curvature zero. Moreover, the ridge curve
crosses the parabolic curve transversely at **P**.

If **X** *A* then the germ
at (**V**, **P**) of is either
regular, a fold, or a cusp. If one of the principal curvatures of
**X** at **P** is zero, then the germ at
(**V**, **P**) of is a fold or a cusp.
If it is a fold, then the germ of ^{~} at
(i(**V**), **P**) is a fold, so **P** is not a ridge
point.

**Theorem 6.3** If **P** is a cusp of the Gauss map of
**X**: *M*^{2} -> **R**^{3},
then given > 0 and
*d* > 0, there exists a point **Q** *U* and a *D* > *d*
such that |**P** - **Q**| < and **Q** is a swallowtail point of the parallel
surface to **X** at distance *D*. If **X** *A* then the cusps of the Gauss map
**X** are the only points of *M* with this property.

**Proof** The parallel map **X**_{r} has a
cusp at (**V**, **Q**) if and only if the map : -> **R**^{3} has a fold at
(*r***V**, **Q**). So a parallel surface has a
cuspidal edge only where it meets the focal surface, and swallowtails
only where it meets the cuspidal edge of the focal surface.

If **P** is a Gaussian cusp, then **P** is a ridge point with
associated principal curvature zero (theorem 6.2). So a point
**Q** near **P** on the ridge curve through **P** has the
desired property. If **X** *A*
then the cusps of the Gauss map are the only points with this
property, by theorem 6.2.

The following corollary implies theorem 3.1(f).

**Corollary 6.4** If **P** is a cusp of the Gauss map of
**X**: *M*^{2} -> **R**^{3},
then for any point **A** in **R**^{3} which is not on
the tangent plane to **X** at **P**, the point **P** is a
swallowtail point of the pedal surface of **X** from **A**. If
**X** *A* then the cusps of the
Gauss map **X** are the only points of *M* with this property.

**Proof** Consider
*f*: *S*^{2} x **R** -> **R**^{3},
*f*(**V**, *t*) = **A** + *t***V**.
Since the Gauss map **N**: -> *S*^{2} is the catastrophe map
of the family : *S*^{2} x *M* -> *S*^{2} x **R**,
the cusps of **N** are the swallowtails of |_{}. But *f*((*M*)) is precisely the pedal surface of **X**
from **A**.

Notice that this result gives a more visual proof of the bitangent
plane characterization of Gaussian cusps, since a swallowtail point of
the pedal surface corresponds to a limit of bitangent planes of
**X**.

The point **P** is a critical point of the distance function
*D*_{A} if and only if the sphere through
**X**(**P**) centered at **A** is tangent to the immersion
**X** at **P**. The point **P** is a degenerate critical
point of *D*_{A} if and only if this sphere
*S*_{A} has unusual contact with **X** at
**P**. For example, if **A** is a focal point of **X** at
**P** then *D*_{A} has Milnor number two at
**P**, and *S*_{A} has stationary contact with
**X** at **P**. As the point **A** goes to infinity in the
direction **V**
*S*^{2}, the sphere *S*_{A}
approaches the plane perpendicular to **V**, and spherical contact
becomes the planar contact discussed at the end of chapter 5.

**Lagrange and Legendre singularities** [A2], [A3], [A4], [AGV], [Wa2]

Lagrangian singularities are generalizations of the singularities
which occur in the normal map : **N** -> **R**^{n+1}
of an immersion
**X**: *M*^{n} -> **R**^{n+1}.
The map is an example of a *Lagrange mapping*,
and the set of critical values of (the envelope of
the family of normal lines of **X**, i.e. the focal surface of
**X**) is its *caustic* [A3].

Recall that the normal map is the catastrophe
map of the family *D* of radial distance-squared functions. In
general, if
*F*: *Q* x *M* -> *Q* x **R**
is a family of real-valued functions on *M* parametrized by
*Q*, then the catastrophe map : *C* -> *Q* is Lagrangian
(cf. [Wa2, lemma 1, p. 716]). The caustic of is
the bifurcation set of the family *F*. In particular, the Gauss
mapping of an immersed hypersurface of Euclidean space is a Lagrange
mapping, since it is the catastrophe map of the family of projections
to lines (Chpater 5). (Cf. [Wa2, prop. 4, p. 720])

Legendre singularities are generalizations of the singularities
which occur in parallel surfaces of an immersed hypersurface of
Euclidean space. The parallel map
**X**_{r}: *M* -> **R**^{n+1}
of the immersion
**X**: *M* -> **R**^{n+1} is
an example of a *Legendre mapping*, and its image (the parallel
hypersurface of **X** at distance *r*) is its *front*
[A2].

If
*F*: *Q* x *M* -> *Q* x **R**
is a family of functions with critical set *C**Q* x *M*, the map
*F*|_{C} is Legendre [A2, thm. 18, p. 33]. For
example, for an immersion
**X**: *M*^{n} -> **R**^{n+1},
consider the family
*F*:*S*^{n} x *M* -> *S*^{n} x **R**
of projections to lines. Choose a point **A** **R**^{n+1} and define
*f*: *S*^{n} x **R** -> **R**^{n+1}
by
*f*(**V**, *t*) = **A** +o*t***V**.
The critical set *C* of *F* is *M*, and
*f*(*F*(*C*)) is the pedal surface (from the point
**A**) of the immersion **X**. Thus the pedal surface of an
immersion has Legendre singularities (away from the point **A**).

A survey of the classification of simple Lagrange and Legendre
singularities is given by Arnold in his paper "Critical points of
smooth functions" [A3]. An excellent description of the geometry and
physics of Lagrange and Legendre singularities is given in Arnold's
book *Mathematical Methods of Classical Mechanics* [A4].

Arnold has communicated to us the following elegant method for
dealing with Gauss maps and dual hypersurface singularities using
Lagrange and Legendre geometry. The set *L* of oriented affine
lines in Euclidean space **R**^{n} is canonically
isomorphic with the symplectic manifold
*T*^{*}*S*^{n-1}, the dual tangent
space of the unit (*n*-1)-sphere. An isomorphism from *L*
to the tangent space *T**S*^{n-1} is
identified with *T*^{*}*S*^{n-1}
using the Euclidean metric
(**V** -> <**V**, ^{.}>). An easy computation
shows that if
**X**: *M*^{n-1} -> **R**^{n}
is an immersion, then the map **N** which assigns to **P** *M* the normal line to **X** at
**P** is a Lagrangian immersion of *M* in *L*. The Gauss
map is the Lagrangian map associated with the Lagrangian immersion
**N**, i.e. the composition of **N** with projection from
*L* = *T*^{*}*S*^{n-1}
to *S*^{n-1}. (Cf. [A4, chapter 8 and appendix
12])

The contractification of the symplectic manifold
T^{*}*S*^{n-1} is the contact manifold
*h* consisting of all hyperplanes (contact elements) in the
tangent bundle of real projective space
**R***P*^{n}. (*H* can be identified with
*P**T*^{*}**R***P*^{n}, the
projectivized cotangent bundle of
**R***P*^{n}.) The Legendre fibration
**N** -> (**R***P*^{n})^{*} (the
dual projective space) assigns to each tangent hyperplane the
(*n*-1)-dimensional projective space to which it is tangent. If
**X**: *M*^{n-1} -> **R***P*^{n}
is an immersion, then the map *T* which assigns to **P** *M* the tangent hyperplane to **X**
at **P** is a Legendre immersion of *M* in *H*. The dual
hypersurface
**X**^{*}: *M* -> (**R***P*^{n})^{*}
is the Legendre map associated with *T*, i.e. the composition of
*T* with the Legendre fibration
**N** -> (**R***P*^{n})^{*}
(cf. [A4, appendix 4]).