THE GEOMETRY OF SHADOW BOUNDARIES ON SURFACES IN SPACE

RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA

                             HOWARD@MATH.SC.EDU

Contents

1. Introduction 1 2. The Second Fundamental Form of a an Oriented Surface. 1 3. The Light Source at an Infinite Distance. 3 4. Light Sources at a Finite Distance. 9 References 11

1. Introduction

These are somewhat expanded notes from a talk on the geometry of shadow curves given in the differential geometry seminar at the University of South Carolina. They are meant to be readable by anyone who has has a one term course in the differential geometry of curves and surfaces, but probably fail at archiving quit that level of accessibility.

2. The Second Fundamental Form of a an Oriented Surface.

Let M^{2} be an oriented surface immersed in R^{3} and let n: M^{2} → S^{2} be the Gauss map (unit normal field) along M^{2}. We denote the standard flat connection on R^{3} by \overline{∇} and the induced connection on M^{2} by ∇. If X and Y are vector fields on M^{2} then these are related by

\overline{∇}_{X}Y = ∇_{X}Y + II(X, Y)n

here II is the second fundamental form of M^{2} which is a symmetric bilinear form. The Weingarten map (or shape operator), A, of M^{2} is a linear map defined on tangent spaces by

A(X) := −\overline{∇}_{X}n.

If ⟨ , ⟩ is the standard inner product on R^{3} then A and II are related by

⟨AX, Y ⟩ = II(X, Y).

Therefore the symmetry of II( ,) implies that A is a self-adjoint linear map. Some idea of the geometric meaning of II( ,) can be form the following:

Exercise 2.1. Let M^{2} be a smooth surface in R^{3} then by a rigid motion we can move M^{2} so that any given point is at the origin and the tangent plane to M at the origin is the x-y plane and that the normal n to M at the origin points upward along the z-axis. Then locally near the origin M^{2} is given by z = f(x, y) where f(0, 0) = f_{x}(0, 0) = f_{y}(0, 0) = 0. Then show the second fundamental form of M^{2} is

II_{(0,0)} = f_{xx}(0, 0) dx^{2} + 2f_{xy}(0, 0) dxdy + f_{yy}(0, 0)dy^{2}.

More generally show that if M^{2} is given as a graph z = f(x, y) over U ⊂ R^{2} the unit normal to M^{2} is

n(x, y) = \frac{1}{\sqrt{1 + f_{x}^{2} + f_{y}^{2}}} [\begin{matrix}−f_{x} \\ −f_{y} \\ 1 \end{matrix}]

and the second fundamental form is given by

II_{(0,0)} = \frac{f_{xx} dx^{2} + 2f_{xy} dxdy + f_{yy} dy^{2}}{\sqrt{1 + f_{x}^{2} + f_{y}^{2}}} .

□

We will also have occasion to work with the covariant derivatives of A and II. If X, Y, Z are smooth vector fields on M then these are defined by

(∇_{X}A)(Y) := ∇_{X}(A(Y)) − A(∇_{X}Y).

and

(∇_{X}II)(Y, Z) := X(II(Y, Z)) − II(∇_{X}Y, Z) − II(Y, ∇_{X}Z).

These are tensors (that is are linear in each X, Y, Z over the smooth functions) and are related by

(∇_{X}II)(Y, Z) = ⟨(∇_{X}A)(Y), Z⟩.

Also

(∇_{X}A)(Y) = (∇_{Y} A)(X)

(This is the Codazzi equation) and (∇_{X}II)(Y, Z) is a symmetric function of all of X, Y, Z.

Exercise 2.2. As in Exercise 2.1 by a rigid motion we can move M^{2} so that any given point is at the origin and the tangent plane to M at the origin is the x-y plane and that the normal n to M at the origin points upward along the z=axis. Then locally near the origin M^{2} is given by z = f(x, y) where f(0, 0) = f_{x}(0, 0) = f_{y}(0, 0) = 0. Then show that at the origin the tensor ∇II is given by

∇II_{(0,0)} = f_{xxx}(0, 0) dx^{3} + 3f_{xxy}(0, 0) dx^{2}dy + 3f_{xyy}(0, 0) dxdy^{2}f_{yyy}(0, 0) dy^{2}.

That is at (0, 0)

(∇_{\frac{∂}{∂x}} II)(\frac{∂}{∂x}, \frac{∂}{∂x}) = f_{xxx}(0, 0),  (∇_{\frac{∂}{∂x}} II)(\frac{∂}{∂x}, \frac{∂}{∂y} ) = f_{xxy}(0, 0),
(∇_{\frac{∂}{∂x}} II)(\frac{∂}{∂y} , \frac{∂}{∂y} ) = f_{xyy}(0, 0),  (∇_{\frac{∂}{∂y}} II)(\frac{∂}{∂y} , \frac{∂}{∂y} ) = f_{yyy}(0, 0).

Therefore if M^{2} is expressed locally as a graph z = f(x, y) as above, then II ∇II give the second and third order terms in the Taylor expanson of M^{2}. □

3. The Light Source at an Infinite Distance.

For any unit vector e in R^{3} define a function f_{e}: M^{2} → R by

(3.1)  f_{e}(x) := ⟨n(x), e⟩.

If we think of a light source at a very great distance from M^{2} then near M^{2} the light rays can be assume parallel (for example light form the sun is generally treated as being parallel). If e is a unit vector parallel to the light rays and pointing toward the light source then the set {x ∈ M^{2} : f)e(x) > 0} can be thought of as the part of M^{2} that is illuminated. (This is not perfect model as it there may be parts of the surface that are illuminated under this definition, but which are in really in the shadow of some other part of the surface.)

The the shadow boundary of M^{2} in the direction of e is the set defined by

S_{e} := {x ∈ M : ⟨n(x), e⟩ = 0}.

That is S_{e} is the zero set of f_{e}. We wish to understand the basic geometry of S_{e}. That is for what e is S_{e} a smooth curve and if S_{e} is not smooth is it at least a graph in the sense of combinatorics, that is S_{e} is a finite set of points connected by smooth curves. The main tool will be the implicit function theorem. We first compute the derivative of f_{e}. For a vector X tangent to M we have

(3.2)  df_{e}(X) = X⟨n, e⟩ = ⟨\overline{∇}_{X}n, e⟩ + ⟨n, \overline{∇}_{X}e⟩ = −⟨A(X), e⟩

as \overline{∇}_{X}e = 0. This leads to

Proposition 3.1. At a point x in the shadow boundary S_{e} the vector e is tangent to M and the derivative of f_{e} is given by

(3.3)  (df_{e})_{x}(X) = −⟨X, Ae⟩ = −II(X, e).

Therefore df_{e} vanishes at the point x of S_{e} if and only if Ae = 0. At points x of M^{2} where Ae ≠ 0 the set S_{e} is a smooth curve in a neighborhood of x.

Proof. At points x ∈ S_{e} we have ⟨n(x), e⟩ = 0 and therefore e ∈ T (M)_{x}. Using the formula (3.2) and that A is self-adjoint we see that (3.3) holds. That (df_{e})_{x} = 0 if and only if Ae = 0 is now clear. If Ae ≠ 0 at x ∈ S_{e} then the zero set of f_{e} is a smooth curve near x by the implicit function theorem. □

Definition 3.2. Two vectors X, Y ∈ T (M)_{x} tangent to a surface at the same point are conjugate iff II(X, Y) = 0. That is vectors are conjugate iff they are orthogonal with respect to the second fundamental form. □

Remark 3.3. If x ∈ S_{e} and Ae ≠ 0 so that S_{e} is a smooth curve near x then let X be tangent to S_{e} at x. Then df_{e}(X) = −II(X, e) = 0. This implies that at its smooth points the shadow boundary is orthogonal, with respect to the second fundamental form, to the direction of the light source. That is X and e are conjugate. We will see below this is also true for light sources at a finite distance. Thus at a point x in a surface M by shining a light on M from directions e tangent to M at x we can see the vectors X conjugate to e as being the tangents to the shadow boundary S_{e} at x. This give quite a bit of information about the second fundamental form because of:

Exercise 3.4. Let b_{1}( ,) and b_{2}( ,) be two symmetric bilinear form on a two dimensional vector space V. Assume that there are three pairwise linear independent vectors v_{1}, v_{2}, v_{3} ∈ V so that

{X ∈ V : b_{1}(X, v_{i}) = 0} = {X ∈ V : b_{2}(X, v_{i}) = 0} for i = 1, 2, 3.

Then show there is a nonzero constant λ so that b_{2} = λb_{1}. □

Therefore three different shadow boundaries through a point allows one to “see” the second fundamental form up to a constant multiple. Some thought, and drawing some pictures, shows that in if φ: R^{3} → R^{3} is an affine map and S_{e} is a shadow boundary on M then φS_{e} will be a shadow boundary on the image φM. This shows that that conformal class of the second fundamental form is an affine invariant of M^{2} ⊂ R^{3}. We will see a generalization (cf. Remark 4.3) of this below. □

On a surface with a Riemannian metric let K denote the Gauss curvature. On an immersed surface in R^{3} this is given by K = det(A) where A is the Weingarten map.

Proposition 3.5. In the space of immersions of a compact oriented surface into R^{3} let K be the set of immersions so that

K = 0 implies dK_{x} ≠ 0.

Then K is an open dense set in the C^{2} in the space of immersions. For all surfaces M^{2} in K the set {K = 0} is a smooth curve in M.

Proof. This follows Thom’s Jet Transversality Theorem [1, Thm 4.9 p. 54]. □

Proposition 3.6. If M ∈ K (with K as defined above) then M has no planar points. That is there are no points x where A_{x} = 0. More precisely let c = {x ∈ M : K(x) = 0}. Then if X ∈ T (M)_{x} is a vector that is not tangent to c then

A_{x} ≠ 0 and ∇_{X}A ≠ 0.

Proof. Let x ∈ c. Then there is a oriented orthonormal moving frame e_{1}, e_{2} defined in a neighborhood x so that for all Y ∈ T (M)_{x} we have (∇_{Y} e_{1})(x) = (∇_{Y} e_{2})(x) = 0. If X ∈ T (M)_{x} and X is not tangent to c then as c is defined by K = 0 we have that dK(X) ≠ 0. This implies that at the point x we have

0 ≠ dK(X)e_{1} ∧ e_{2} = ∇_{X}(Ke_{1} ∧ e_{2})
= ((∇_{X}A)e_{1}) ∧ (Ae_{2})) + (Ae_{2}) ∧ ((∇_{X}A)e_{2}).

Now A_{x} ≠ 0 and ∇_{X}A ≠ 0 follows at once. □

For a smooth surface M^{2} immersed in R^{3} recall that we classify the points x ∈ M as follows:

⇒ The point x is elliptic iff K(x) > 0. This means that II( ,) is either positive or negative definite. Near such a point the M is locally convex in the sense that if locally lies on one side of its tangent plane. The ellipsoids x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1 are examples of surfaces with all points elliptic.

⇒ The point x is hyperbolic iff K(x) < 0. In this case II( ,) is nondegenerate and indefinite. In this case M is locally a saddle surface near x. The surface z = xy and the hyperboloid of one sheet x^{2}/a^{2} + y^{2}/b^{2} − z^{2}/c^{2} = 1 are example of surfaces with all points hyperbolic. ⇒ The point x is parabolic iff K(x) = 0 but II_{x}( ,) ≠ 0. In this case A_{x} has and one zero eigenvalue and one nonzero eigenvalue. The cylinder x^{2}/a^{2} ± y^{2}/b^{2} = 1 is a surface with all points parabolic. ⇒ The point x is planar iff II_{x}( ,) = 0. The surface z = xy(x − y) has a planer point at the origin. A connected surface with all its points planar is an open subset of a plane.

The set of all parabolic points of M^{2} is call the parabolic locus of M^{2}. If c is the parabolic locus of M^{2} then Propositions 3.7 and 3.6 taken together imply that if M^{2} ∈ K that the parabolic locus in M^{2} is defined by {K = 0} and that this is a smooth curve in M.

Now let M^{2} ∈ K and let S(M) be the unit sphere bundle of M. Then for each x ∈ M^{2} with K(x) = 0 there are exactly two unit vectors u_{1}, u_{2} ∈ T (M)_{x} so that Au_{1} = Au_{2} = 0. (This is because A ≠ 0 and det A = 0 implies A has rank one.) Let

(3.4)  B := {(x, u) ∈ S(M) : K(x) = 0, u ∈ T (M)_{x}, and Au = 0}.

The as {K = 0} is a smooth curve and B double covers {K = 0} the curve {K = 0} we have that B is a smooth curve in S(M). There is a natural smooth map g: B → S^{2} given by

g(x, u) = u.

Proposition 3.7. With notation as above if M^{2} ∈ K and e ∈ S^{2} \ g[B] then the shadow boundary S_{e} is a smooth imbedded one dimensional submanifold of M^{2}. (Note that g[B] is a smooth image of a smooth curve so that g[B] not only has two dimensional measure zero, but is a closed set of finite length, i.e. one dimensional Hausdorff measure. Thus for almost all e the shadow boundary S_{e} will be smooth.)

Proof. Let x ∈ S_{e}. By Proposition 3.1 it is enough to show that for all x ∈ S_{e} that A_{x}e ≠ 0. But if A_{x}e = 0 then (x, e) ∈ B which would in turn imply that e ∈ g[B] which is not the case. □

Definition 3.8. Let V be a finite set of points on the surface. Then by a generalized graph on M with vertices V we mean a subset c of M so that V ⊆ c and c \ V is a smooth embedded curve in M \ V.

Proposition 3.9. Let M^{2} ∈ K and let e ∈ S^{2} so that the set

V := {x ∈ M^{2} : A_{x}e = 0}

is finite. Then the shadow boundary S_{e} is a generalized graph on V. In particular this implies that in some generalized sense S_{e} is a piecewise smooth curve.

Proof. This follows at once from Proposition 3.1. □

We compute the Hessian of f_{e}. Let X, Y be smooth vector fields on M^{2} and f: M → R a smooth function. Then the Hessian of D^{2}f is defined by

D^{2}f(X, Y) := Xdf(Y) − df(∇_{X}Y).

As a function of X this is clearly linear over the smooth functions. Using the the connection ∇ is symmetric, that is ∇_{X}Y − ∇_{Y} X = [X, Y], we have

D^{2}f(X, Y) − D^{2}f(Y, X) = Xdf(Y) − Y df(X) − df(∇_{X}Y − ∇_{Y} X)
= [X, Y]f − df([X, Y])
= [X, Y]f − [X, Y]f = 0.

Therefore D^{2}f(X, Y) is symmetric in X and Y. Thus D^{2}f(X, Y) is also linear as a function of Y over the smooth functions. Thus D^{2}f( ,) is a tensor. Note the at a critical point x of f we have df_{x}(∇_{X}Y) = 0 and thus

(3.5)  D^{2}f(X, Y) = Xdf(Y) = XY f at points x where df_{x} = 0.

Exercise 3.10. Let x, y be local coordinates on M whose domain contains the point p. Let f: M → R be a smooth function function that has a critical point at p. Then use (3.5) to show that at p the Hessian is

D^{2}f = f_{xx} dx^{2} + 2f_{xy} dxdy + f_{yy} dy^{2}

at the point p. Therefore at critical points the Hessian is just the usual Hessian we know and love from advanced calculus. (At points other than critical points the local coordinate form of the Hessian will involve the Christoffel symbols Γ_{ij}^{k} of the connection.) □

Definition 3.11. Let f: M^{2} → R be a smooth function. Then a critical point p of f is non-degenerate iff the Hessian D^{2}f_{p}( ,) is a non-degenerate bilinear form. The function f is a Morse function iff is a all the critical points of f are non-degenerate.

Morse functions are particular nice because of the following result due to M. Morse (which also holds in higher dimensions).

Proposition 3.12 (The Morse Lemma). Let p be a non-degenerate critical point of f: M^{2} → R. Then there is a local coordinate system x, y centered at p so that

(1) If the Hessian D^{2}f_{p}( ,) is positive definite then

f = f(p) + x^{2} + y^{2}

in a neighborhood of p

(2) If the Hessian D^{2}f_{p}( ,) is negative definite then

f = f(p) − x^{2} − y^{2}

in a neighborhood of p

(3) If the Hessian D^{2}f_{p}( ,) is indefinite then

f = f(p) + x^{2} − y^{2} = f(p) + (x + y)(x − y)

in a neighborhood of p

Proof. See [1, Thm 6.9 p. 65]. □

We now commute the Hessian of the functions f_{e} given by (3.1).

D^{2}f_{e}(X, Y) = Xdf_{e}(Y) − df_{e}(∇_{X}Y)
= −(X⟨AY, e⟩ − ⟨A(∇_{X}Y), e⟩)
= −(⟨\overline{∇}_{X}(AY), e⟩ + ⟨AY, \overline{∇}_{X}e⟩ − ⟨A(∇_{X}Y), e⟩)
= −(⟨∇_{X}(AY) + II(X, Y)n, e⟩ − ⟨A(∇_{X}Y), e⟩)
= −(⟨∇_{X}(AY) − A(∇_{X}Y)e⟩ + ⟨II(X, Y)n, e⟩)

= −⟨(∇_{X}A)(Y), e⟩ − II(X, Y)⟨n, e⟩ (3.6)

This leads to

Proposition 3.13. Let p be a point where f_{e}(p) = 0 (so that p is on the shadow boundary S_{e}). Assume that p is a critical point of f_{e}, then the Hessian of f_{e} is given by

(3.7)  D^{2}f_{e}(X, Y) = −⟨(∇_{X}A)(Y), e⟩ = −(∇_{e}II)(X, Y).

Therefore the point p is a non-degenerate critical point if and only if the symmetric bilinear form (∇_{e}II)(X, Y) is non-degenerate.

Proof. At a point of where f_{e} = 0 we have ⟨n, e⟩ = 0 and so (3.6) reduces to (3.7). □

As a concrete example let M^{2} be the graph of z = cx^{2}/2 + q(x, y) where q(x, y) is a function that vanishes to third order at the origin. That is q(0, 0) = q_{x}(0, 0) = q_{y}(0, 0) = q_{xx}(0, 0) = q_{xy}(0, 0) = q_{yy}(0, 0). Then the second fundamental form of M^{2} at the origin is

II = cdx^{2}.

and so if c ≠ 0 then the origin is a parabolic point of M^{2}. To see what the parabolic locus looks like near the origin recall that for a graph z = φ(x, y) the Gauss curvature is given by

K(x, y) = \frac{φ_{xx}φ_{yy}^{2} − φ_{yy}^{2}}{(1 + φ_{x}^{2} + φ_{y}^{2})^{2}} .

See for example [2, p. 95]. In our case this becomes

K = \frac{q_{yy}(c + q_{xx}) − q_{xy}^{2}}{(1 + (cx + q_{x})^{2} + q_{y}^{2})^{2}} .

As q vanishes to order three the function q_{x} and q_{y} vanish to order two and q_{xx}, q_{xy}, and q_{yy} vanish to order one. Thus we see that

K = cq_{yy} + O(2)

where O(2) represents terms that vanish to order at least two. Thus if c =≠ 0 and either q_{yyx}(0, 0) ≠ 0 or q_{yyy}(0, 0) ≠ 0 (that is (dq_{yy})_{(0,0)} ≠ 0) then we have that K(0, 0) = 0 and dK_{(0,0)} ≠ 0.

Let e be in the direction of the positive y axis then a calculation shows that

f_{e} = \frac{−q_{y}}{\sqrt{1 + (cx + q_{x})^{2} + (q_{y})^{2}}} .

(This is just the y-component of the normal.) As q vanishes to order 3 q_{y} vanishes to order 2 and so the origin is a critical point of f_{e}. Then by messy calculation we

(D^{2}f_{e})_{(0,0)} = −(q_{xxy} dx^{2} + 2q_{xyy} dxdy + q_{yyy} dy^{2}).

(I confess to having used Maple to do this calculation.)

This can be summarized as

Lemma 3.14. If e = (0, 1, 0) and q vanishes to third order at the origin and

(1) c ≠ 0,

(2) q_{yyx}^{2}(0, 0) + q_{yyy}^{2} ≠ 0,

(3) q_{xxy}(0, 0)q_{yyy}(0, 0) − q_{xyy}^{2}(0, 0) ≠ 0. Then

(1) K(0, 0) = 0,

(2) dK_{(0,0)} ≠ 0,

(3) Ae = 0

(4) The function f_{e} has a non-degenerate critical point at the origin.

The conditions on functions q vanishing to order three at the origin are open and dense in the C^{3} topology on such functions.

Definition 3.15. Let N be the set of all compact oriented surfaces M^{2} immersed in R^{3} so that

(1) M^{2} ∈ K

(2) The number of points p in the parabolic locus {K = 0} of M^{2} with ∇_{e}II(X, Y) a degenerate quadratic form (where e is a unit vector with A_{p}e = 0) is finite. □

Proposition 3.16. The set N is open and dense in the C^{3} topology on the space of immersions. If M^{2} ∈ N then for all but finitely many vectors e we have that either

(1) 0 is a regular value of f_{e} so that the shadow boundary S_{e} is a smooth curve, or

(2) any critical point p of f_{e} on the shadow boundary S_{e} is a nondegenerate critical point and thus by the Morse Lemma 3.12 we have a norm form for f_{e} near p (and this also a normal form for the shadow boundary S_{e} near p).

Proof. That N is open and dense in the C^{3} topology follows from the calculations of Lemma 3.14 and Thom’s Jet Transversality Theorem [1, Thm 4.9 p. 54].

If M^{2} is in N and let x_{1}, …, x_{n} be the finite number of points x on the parabolic locus c := {K = 0} where (∇_{v}II)_{x}( ,) is degenerate (here v is a unit vector so that A_{x}v = 0). Let v_{1}, …, v_{n} be unit vectors so that v_{i} is tangent to M at x_{i} and A_{x_{i}}v_{i} = 0. Then if e is a unit vector so that f_{e} has a degenerate critical point x on the shadow boundary S_{e} then by Proposition 3.1 the point x is on {K = 0} and A_{x}e = 0. By Proposition 3.13 as the Hessian of f_{e} is degenerate at x we have that x = x_{i} for some i and that e = ±v_{i}. Therefore the only directions where f_{e} has a degenerate critical point on S_{e} is when e is one of the 2n vectors ±v_{1}, …, ±v_{n}. This completes the proof. □

4. Light Sources at a Finite Distance.

Let a ∈ R^{3} and for a smooth oriented surface M^{2} in R^{3} define F_{a}: M^{2} → R^{3} by

F_{a}(x) := ⟨n(x), a − x⟩.

Then the set {x ∈ M^{2} : F_{a}(x) > 0} can be thought of as the set of points on M^{2} that are illuminated be a light source at a. In this case we define the shadow boundary by

S_{a} := {x : F_{a}(x) = 0}.

As in the case of a light source coming from infinity we wish to investigate when S_{a} is a well behaved subset of M^{2}. We first compute the derivative of F_{a}. Let X be tangent to M^{2}. Then

dF_{a}(X) = X⟨n, a − x⟩ = ⟨\overline{∇}_{X}n, a − x⟩ + ⟨n, \overline{∇}_{X}(a − x)⟩
= ⟨−AX, a − x⟩ + ⟨n, −X⟩
= ⟨AX, x − a⟩

where we have used that \overline{∇}_{X}a = 0 and \overline{∇}_{X}x = X. This leads to

Proposition 4.1. Let p ∈ S_{a}. Then

(dF_{a})_{p}(X) = ⟨X, A(p − a)⟩ = II(X, p − a⟩.

Therefore at a point p of the shadow boundary S_{a} we have (dF_{a})_{p} = 0 if and only if A(p − a) = 0. If A(p − a) ≠ 0 then S_{a} is a smooth curve near p.

Exercise 4.2. Give a proof of this along the lines of Proposition 3.1. □

Remark 4.3. This shows that at smooth points p of S_{a} the tangent space to S_{a} is defined by {X ∈ T (M)_{p} : II(X, p − a) = 0}. Therefore, just as in the case of the light source at infinity, the tangent to the shadow boundary and the direction of the light source are conjugate directions. Then as in Remark 3.3 this can be used to show that the conformal class of the second fundamental form is invariant under projective transformations. □

Define the class of immersed surfaces K as above and let B be defined by (3.4). Then given M^{2} ∈ K let L be

L := {p + tu : (p, u) ∈ B, t ∈ R}.

This is just the union of the lines t ↦ p + tu with (p, u) ∈ B. As B is a smooth curve the set L is a ruled surface in R^{3} (which in general will have a non-empty singular set).

Proposition 4.4. Let M^{2} ∈ M. Then for all a ∈ R^{3} \(L∪M^{2}) the shadow boundary S_{a} is a smooth embedded curve in M^{2}. Thus for almost all a ∈ R^{3} the shadow boundary S_{a} is a smooth curve.

Exercise 4.5. Prove this Proposition. Hint: Modify the proof of Proposition 3.7. □

Proposition 4.6. Let M^{2} ∈ K and let a ∈ R^{3}\ so that the set

V := {p ∈ M^{2} : A_{p}(p − a) = 0}

is finite. Then the shadow boundary S_{e} is a generalized graph on V.

Exercise 4.7. Prove this. Hint: See the proof of Proposition 3.9 □

Exercise 4.8. Show the Hessian of F_{a}: M^{2} → R^{3} is given by

(D^{2}F_{a})_{p}(X, Y) = −II(X, Y)F_{a}(p) + (∇_{p−a})(X, Y) + II(X, Y)

for any p ∈ M^{2}. If p ∈ S_{a} = {F_{a} = 0} show this simplifies to

(D^{2}F_{a})_{p}(X, Y) = (∇_{p−a}II)(X, Y) + II(X, Y).

Therefore if p ∈ S_{a} is a critical point of F_{a} (that is A_{p}(p − a) = 0) then p is a non-degenerate critical point if and only if the quadratic form

(X, Y) ↦ (D^{2}F_{a})_{p}(X, Y) = (∇_{p−a}II)(X, Y) + II(X, Y)

is non-degenerate. □

Exercise 4.9. Let b_{1}( ,) and b_{2}( ,) be two symmetric bilinear forms on a two dimensional vector space V. Assume that b_{1} has rank one and b_{2} is nondegenerate. Then there is exactly two real numbers so that the quadratic form b_{1}( ,) − tb_{2}( ,) is degenerate and one of these values if t = 0. (It is possible that t = 0 with multiplicity two and that there are no non-zero t with tb_{2} + b_{1} degenerate). Thus there is at most one non-zero value of t so that b_{1}( ,) + tb_{2}( ,) is degenerate. Hint: Let P (t) = det(b_{1} − tb_{2}). Then b_{1}( ,)−tb_{2}( ,) is degenerate if and only if P (t) = 0. Clearly P (0) = det(b_{1}) = 0 as b_{1} has rank one.

Proposition 4.10. Let M^{2} ∈ N (where N is as in Definition 3.15). For any (p, u) ∈ B (with B given by (3.4)) let L_{p} be the line

L_{p} := {p + tu : t ∈ R}.

Then for all but finitely many points p ∈ {K = 0} there are only two point a ∈ L_{p} so that p is a degenerate critical point of F_{a} and one of these points is a = p.

Exercise 4.11. Prove this. Hint: Any point a on the line P_{p} is of the form a = p + tu and so p − a = −tu. But then for X, Y ∈ T (M)_{p} we have

(D^{2}F_{a})_{p}(X, Y) = (∇_{p−a}II)(X, Y) + II(X, Y) = −t(∇_{u}II)_{p}(X, Y) + II_{p}(X, Y).

The rank of II_{p}( ,) is one by Proposition 3.6 and (∇_{u}II)_{p}( ,) is non-degenerate for all but finitely many p ∈ {K = 0} as M^{2} ∈ N. Now use Exercise 4.9. □

References

1. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Springer-Verlag, New-York, 1973.

2. M. Spivak, A comprehensive introduction to differential geometry, 2 ed., vol. 2, Publish or Perish Inc., Berkeley, 1979.