MOHAMMAD GHOMI’S SOLUTION TO THE ILLUMINATION PROBLEM

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

                           HOWARD@MATH.SC.EDU

Contents

1. Introduction 1 2. Elementary Results about the Boundaries of Shadows 2 3. An Application of Chern-Lashof Theory 3 4. Nicely Placed Triples in Oriented Surfaces 5 5. Proof of the Theorem 8 6. Tubes about Curves and an Example of a Genus One Surface

with Connected Shadows 11 7. Appendix: A Sufficient Condition for a Spherical Curve to be the

Tangent Mapping of a Space Curve 14 References 15

1. Introduction

Let M^{2} be a compact oriented surface immersed in R^{3} with unit normal field n. Recently Mohammad Ghomi has proven the following very pretty result.

Theorem. If for all unit vectors e the sets {x ∈ M^{2} : ⟨n(x), e⟩ > 0} are simply connected, then M^{2} is the boundary of a bounded convex set in R^{3}.

This problem has been around for a while in the from of the “illumination conjecture”. Ghomi also gave an example to show that “simply connected” can not be weakened to “connected” as there is an imbedded torus M^{2} in R^{3} so that all of the sets {x ∈ M^{2} : ⟨n(x), e⟩ > 0} are connected (see Figure 7).

Below is my exposition of Ghomi’s results. Any errors are mine and not his.

Date: December 1998 with Sections 6 and 7 added in April 1999. Minor corrections added October 2003.

2. Elementary Results about the Boundaries of Shadows

Let M^{2} be a compact oriented surface immersed in R^{3}. The orientation of M^{2} determines a unit normal field n along M^{2} in the usual manner. We now fix our notation for the imbedding invariants of M^{2} in R^{3}. Let \overline{∇} be the usual flat connection on R^{3}. Then the second fundamental form II and the induced connection ∇ on M^{2} are defined by

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

where X and Y are smooth vector fields defined on M^{2} and ∇_{X}Y is the component of \overline{∇}_{X}Y tangent to M^{2}. The Weingarten map A (or shape operator) is the linear map defined on vectors tangent to M^{2} by

AX := −\overline{∇}_{X}n

and is related to the second fundamental form by ⟨AX, Y ⟩ = II(X, Y).

Let S^{2} be the unit sphere in R^{3} and for each e ∈ S^{2} define a function f_{e}: M^{2} → R by

f_{e}(x) := ⟨n(x), e⟩.

Then the shadow set defined by e is

S_{e} := {x ∈ M^{2} : f_{e}(x) > 0}.

The boundary ∂S_{e} is the shadow boundary in the direction e. We now show that for most choices of e ∈ S^{2} the shadow boundary is a smooth curve.

Proposition 2.1. For almost all e ∈ S^{2} (in the sense of Lebesgue measure on S^{2}) the set {x ∈ M^{2} : f_{e}(x) = 0} is a smooth curve in M^{2}. Thus for these e we have that ∂S_{e} = {x ∈ M^{2} : f_{e}(x) = 0} and so ∂S_{e} is a smooth curve for almost all e ∈ S^{2}.

Proof. Let U(M) := {(x, u) : x ∈ M^{2}, u ∈ T (M)_{x}, ∥u∥ = 1} be the unit sphere bundle of M^{2}. Define a map τ: U(M) → S^{2} by τ(x, u) = u (where u is translated to the origin). By Sard’s Theorem almost every e ∈ S^{2} is a regular value of τ and for such e the preimage τ^{−1}[e] is a smooth curve in U(M). If (x, u) ∈ U(M) and u_{1} ∈ T (M) is a unit vector with ⟨u, u_{1}⟩ = 0, then c(t) = (x, cos(t)u+sin(t)u_{1}) parameterizes the fiber U(M)_{x} and c(0) = (x, u). Also

τ_{∗}c′(0) = \frac{d}{dt}τ(x, cos(t)u + sin(t)u_{1})|_{t=0} = \frac{d}{dt}(cos(t)u + sin(t)u_{1})|_{t=0} = u_{1}

and so τ_{∗}c′(0) ≠ 0. If e is a regular value of τ and (x, u) ∈ τ^{−1}[e] then the tangent space to τ^{−1}[e] is {X ∈ T (U(M))_{(x,u)} : τ_{∗}X = 0}. As τ_{∗}c′(0) ≠ 0 this implies that the curve τ^{−1}[e] is never tangent to any of the fibers U(M)_{x}.

Let π: U(M) → M^{2} be the natural projection and e a regular value of τ. Then a chase through the definitions involved shows the map π|_{τ^{−1}[e]} : τ^{−1}[e] → S^{2} is injective. Because τ^{−1}[e] is never tangent to the fibers π^{−1}[x] = U(M)_{x} the map π|_{τ^{−1}[e]} is an immersion. As τ^{−1}[e] is compact this implies that the projection π[τ^{−1}[e]] is a smooth embedded curve in M^{2}. To finish the proof note that {f_{e} = 0} is equal to this projection:

π[τ^{−1}[e]] = {x ∈ M^{2} : e ∈ T (M)_{x}} = {x ∈ M^{2} : ⟨e, n(x)⟩ = 0}

= {x ∈ M^{2} : f_{e}(x) = 0}. □

Corollary 2.2. Let M^{2} be a compact oriented surface immersed in R^{3} so that all the shadow sets S_{e} are simply connected. Then M^{2} is diffeomorphic to the sphere.

Proof. As in the last proof choose e to be a regular value of τ. Then ∂S_{e} is a smooth curve. As S_{e} is simply connected the closure \overline{S}_{e} will be diffeomorphic to a closed disk. The boundary of S_{−e} is same as the boundary of S_{e} and so \overline{S}_{−e} is also a closed disk. The surface M^{2} is the disjoint union of S_{e}, S_{−e} and ∂S_{e}. Therefore M^{2} is a pair of disks glued together along their boundaries and thus M^{2} is a sphere. □

If is possible to be more precise about at which points the shadow boundaries ∂S_{e} are smooth curve. We first compute the differential of f_{e}. For a vector X tangent to M we have

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

as \overline{∇}_{X}e = 0. If X is tangent to M at a point x ∈ ∂S_{e} then f_{e}(x) = ⟨e, n(x)⟩ = 0 and so e ∈ T (M)_{x}. Thus at points x ∈ ∂S_{e} we can use that A is selfadjoint to write

df_{e}(X) = −⟨X, Ae⟩.

This leads to:

Proposition 2.3. If x ∈ ∂S_{e} and Ae ≠ 0 then ∂S_{e} is a smooth curve near x. In particular if the Gaussian curvature K = det(A) does not vanish at x then ∂S_{e} is a smooth curve near x for all e⊥n(x).

Proof. If Ae ≠ 0 then df_{e} = −⟨·, A(e)⟩ is a not zero at x and thus {f_{e} = 0} is a smooth curve by the implicit function theorem. At points x where K ≠ 0 we have Ae ≠ 0 for all e ≠ 0 and e⊥n(x) (which is the condition that e ∈ T (M)_{x}). □

3. An Application of Chern-Lashof Theory

We recall the basics of the Chern-Lashof theory as applied to surfaces in R^{3} (the original results work in all dimensions and codimensions). Let M^{2} be a compact oriented surface immersed in R^{3}. For any unit vector u ∈ S^{2} define the height function h_{u}: M^{2} → R in the direction u as h_{u}(x) := ⟨x, u⟩. Recall that if n is the unit normal field along M^{2} then the Gauss map of M^{2} is the function g: M^{2} → S^{2} given by g(x) = n(x) translated to the origin. Also recall that a function is a Morse function iff all of its critical points are non-degenerate. That is at any critical point the Hessian is non-singular.

Proposition 3.1. If u ∈ S^{2} and both u and −u are regular values of the Gauss map then the height function h_{u} is a Morse function on M^{2}. For any such height function the Gauss curvature is non-zero at all critical points of h_{u}. Therefore by Sard’s Theorem h_{u} is a Morse function for almost all u ∈ S^{2} so that the Gauss curvature is non-zero at all critical points. As M^{2} is compact the set of such u is an open set (as the set of critical values is the continuous image of the set of critical points which is compact).

Proof. The derivative of the Gauss map is g_{∗x}X = −AX (where we are identifying the tangent spaces T (M)_{x} and T (S^{2})_{g(x)} by parallel translation) and so x is a critical point of g if and only if det(A_{x}) = 0. But the Gauss curvature is K(x) = det(A_{x}) and thus the set of critical points of g is just the set where the Gauss curvature vanishes. Now let u be a regular value of g. Then any x with g(x) = u is a regular point of g and so K(x) ≠ 0. The differential of h_{u} is dh_{u}(X) = ⟨X, u⟩ and so the critical points of the height function h_{u} are exactly the points x with g(x) = ±u. Let x be a critical point of h_{u}, so that u = ±n, and let X, Y be smooth vector fields on M^{2} so that ∇X = ∇Y at x. Then we can compute the Hessian of h_{u} at x:

D^{2}h_{u}(X, Y) = Xdh_{u}(Y) = X⟨Y, u⟩ = ⟨\overline{∇}_{X}Y, u⟩
= ⟨∇_{X}Y + II(X, Y)n, u⟩ = ±II(X, Y) = ±⟨AX, Y ⟩.

But as det(A_{x}) = K(x) ≠ 0 this implies the Hessian is non-degenerate. □

If u ∈ S^{2} is so that h_{u} is a Morse function on M^{2} then let C(h_{u}) := {x ∈ M^{2} : (dh_{u})_{x} = 0} be the set of critical points of h_{u} (which will be a finite set when h_{u} is a Morse function). Let #C(h_{u}) be the number of points in C(h_{u}). Then the following is implicit in the paper [2] of Chern and Lashof.

Theorem 3.2 (Chern-Lashof [1, 2]). If M^{2} is a compact oriented surface immersed in R^{3} then

(3.1)  \frac{1}{2} ∫_{S^{2}} #C(h_{u}) du = ∫_{M^{2}} |K(x)| dx.

(Here du is the area measure on S^{2} and dx is the area measure on M^{2}.) Note that for almost all u ∈ S^{2} the function h_{u} is a Morse function and thus #C(h_{u}) is finite. Since every smooth function on M^{2} has at least two critical points (corresponding to the maximum and minimum) the inequality

∫_{M^{2}} |K(x)| dx ≥ 4π

holds. Equality holds in this inequality if and only if M^{2} is the boundary of a bounded convex set. □

Let f: M^{2} → R be a Morse function. If x is a critical point of f the index of f at x is defined to be in index of the Hessian D^{2}f_{x} at x. (The index of a symmetric matrix is the number of negative eigenvalues. The index of a symmetric bilinear form is the index of any matrix that represents it.) Let ι_{k}(f) be the number of critical points of f of index k. Then ι_{0}(f) is the number of local minima of f, ι_{2}(f) is the number of local maxima of f and ι_{1}(M) is the number of saddle points of f. Let β_{k}(M^{2}) = dim H_{k}(M, R) be the k-th Betti number of M^{2}. Then β_{0}(M^{2}) = β_{2}(M^{2}) = 1 and β_{1}(M^{2}) = 2g where g is the genus of M^{2}. Then basic results in Morse theory [3] imply

ι_{0}(f) − ι_{1}(f) + ι_{2}(f) = 2 − 2g
ι_{0}(f), ι_{2}(f) ≥ 1
ι_{1}(f) ≥ β_{1}(M^{2}) = 2g.

As an aside we note these inequalities yield that for a surface of genus g that for any Morse function f: M^{2} → R that the number of critical points of f satisfies

#C(f) = ι_{0}(f) + ι_{1}(f) + ι_{2}(f) ≥ 2 + 2g.

Using this with the Chern-Lashof theorem yields

∫_{M^{2}} |K(x)| dx ≥ 4π(1 + g)

for compact oriented surfaces of genus g immersed in R^{3}.

If M^{2} is a sphere so that g = 0 then for any Morse function f on M^{2} we have ι_{0}(f) − ι_{1}(f) + ι_{2}(f) = 2. Thus if f has only one local maximum and one local minimum then ι_{1}(f) = 0. The total number of critical points is then #C(f) = ι_{0}(f) + ι_{1}(f) + ι_{2}(f) = 2. This observation can be combined with the Chern-Lashof theorem to give the following (which can be proven by more elementary methods, but I like Chern-Lashof theory).

Proposition 3.3. Let M^{2} be a sphere immersed in R^{3} in such a way that for almost all u ∈ S^{2} the height function h_{u} has only one local maximum. Then M^{2} is the boundary of a bounded convex set in R^{3}.

Proof. As a local minimum of h_{u} is a local maximum of h_{−u} for almost all u ∈ S^{2} the function will have at most one local maximum and at most one local minimum and thus exactly one of each. As above this implies #C(h_{u}) = 2. But using #C(h_{u}) = 2 for almost all u in (3.1) gives ∫_{M^{2}} |K(x)| dx = 4π. Then Theorem 3.2 implies that M^{2} is the boundary of a convex set. This completes the proof. □

4. Nicely Placed Triples in Oriented Surfaces

We now study the elementary combinatorics of triples of points on the boundaries of simply connected sets in compact surfaces.

Definition 4.1. Let M^{2} be a compact oriented surface. Then (p_{1}, p_{2}, p_{3}, U) is a nicely placed triple in M^{2} iff

(1) U is a simply connected open set in M^{2},

(2) p_{1}, p_{2}, and p_{3} are distinct points on the boundary, ∂U, of U, and (3) for each i = 1, 2, 3 there is an open neighborhood V_{i} of p_{i} in M^{2} so

that V_{i} ∩ ∂U is a smooth curve. □

Note that while the boundary of U in a nicely placed triple (p_{1}, p_{2}, p_{3}, U) is smooth near each of the points p_{i}, there is no regularity assumption on other parts of the boundary. (See Figure 1).

p_{3}  p^{1}  p_{2}

Figure 1. A nicely placed triple (p_{1}, p_{2}, p_{3}, U). The boundary, ∂U, of U is smooth near p_{1}, p_{2}, and p_{3}, but no regularity is required on the rest of ∂U.

If (p_{1}, p_{2}, p_{3}, U) is a nicely placed triple then for any pair of the points p_{1}, p_{2}, p_{3}, to be definite say the pair p_{1} and p_{2}, a curve γ: [0, 1] → M^{2} is a standard curve iff γ is a smooth embedded curve with γ(0) = p_{1}, γ(1) = p_{2} and γ(t) ∈ U for all t ∈ (0, 1). As U is simply connected any two standard curves γ, γ̃: [0, 1] → M^{2} from p_{1} to p_{2} will be homotopic to each other through a family of standard curves. Also the curve γ will divide U into two simply connected regions, one on each side of γ.

Start with a standard curve γ_{12} : [0, 1] → M^{2} from p_{1} to p_{2}. Now p_{3} will lie in the closure of one of the connected components of U \ γ_{12}[(0, 1)]. As this component is simply connected there is a standard curve γ_{2,3} : [0, 1] → M^{2} connecting p_{2} to p_{3} and which is disjoint from γ_{12} except at p_{2}. Then U \ (γ_{12}[(0, 1)] ∪ γ_{23}[(0, 1)]) has three simply connected componets and only one of them has both p_{1} and p_{3} in its closure. Then there is a standard curve γ_{31} : [0, 1] → M^{2} from p_{3} to p_{1} and so that γ_{31} is disjoint from both of γ_{12} and γ_{23} excepty at the points p_{1} and p_{3}. Call such a triple of curves γ_{12}, γ_{23} and γ_{31} a standard triangle for (p_{1}, p_{2}, p_{3}, U). (Figure 2 shows a standard triangle for (p_{1}, p_{2}, p_{3}, U).) Again using that U is simply connected if γ̃_{12}, γ̃_{23} and γ̃_{31} is another standard triangle for (p_{1}, p_{2}, p_{3}, U) then γ̃_{12}, γ̃_{23}, γ̃_{31} is homotopic to γ_{12}, γ_{23}, γ_{31} through standard triangles. That is standard triangles are unique up to homotopy.

Any standard triangle γ_{12}, γ_{23}, γ_{31} for (p_{1}, p_{2}, p_{3}, U) bounds a simply connected region, Int(γ_{12}, γ_{23}, γ_{31}), of U which can be defined as follows. Let U_{12} be the component of U \ γ_{12}[(0, 1)] that contains p_{3} in its closure, U_{23} the component of U \ γ_{23}[(0, 1)] that contains p_{1} in its closure, and U_{31} the component of U \ γ_{31}[(0, 1)] that contains p_{2} in its closure. Then Int(γ_{12}, γ_{23}, γ_{31}) = U_{12} ∩ U_{23} ∩ U_{31}. (If we wish to indicate the dependence on the set U we will write Int_{U}(γ_{12}, γ_{23}, γ_{31}).) Then Int(γ_{12}, γ_{23}, γ_{31}) has

p_{3}  p_{2} p^{1} γ_{12} γ_{31} γ_{23}

Figure 2. This shows a standard triangle for the nicely place triple (p_{1}, p_{2}, p_{2}, U). As U is simply connected any pair of standard triangles for (p_{1}, p_{2}, p_{2}, U) will be homotopic.

a piecewise smooth boundary and as M^{2} is oriented there is an induced boundary orientation on ∂ Int(γ_{12}, γ_{23}, γ_{31}) = γ_{12} ∪ γ_{23} ∪ γ_{31}. (Figure 3 shows one of the two possible boundary orintations on ∂ Int(γ_{12}, γ_{23}, γ_{31}).) This boundary orientation determines a premutation, σ = σ(p_{1}, p_{2}, p_{3}, U) of the set {p_{1}, p_{2}, p_{3}} in a natural manner. That is if when moving along ∂ Int(γ_{12}, γ_{23}, γ_{31}) in the direction of the orientation when leaving p_{i} the next of the points encountered is p_{j} then σ(p_{i}) = p_{j}. (In Figure 3 with the pictured boundary orientation we have (using the standard cycle notation for permutations) σ(p_{1}, p_{2}, p_{3}, U) = (p_{1}p_{2}p_{3}). If the boundary orientation were reversed then σ(p_{1}, p_{2}, p_{3}, U) = (p_{1}p_{3}p_{2}).) As standard triangles are unique up to homotopy it follows that σ(p_{1}, p_{2}, p_{3}, U) is independent of the choice of the standard triangle used to define it. For future use we record the following elementary result whose proof is left to the reader.

Lemma 4.2. If (p_{1}, p_{2}, p_{3}, U) is a standard triple and ∂U is a smooth curve, then σ(p_{1}, p_{2}, p_{3}, U) is the permutation of {p_{1}, p_{2}, p_{3}} defined by moving along ∂U with the induced boundary orientation. □

Let (p_{1}, p_{2}, p_{3}, U) be a standard triple then another standard triple (p_{1}, p_{2}, p_{3}, Ũ) is adjacent to (p_{1}, p_{2}, p_{3}, U) iff there is some standard triangle γ_{12}, γ_{23}, γ_{31} for (p_{1}, p_{2}, p_{3}, U) so that γ_{12}, γ_{23}, and γ_{31} are also standard curves in Ũ and Int_{U}(γ_{12}, γ_{23}, γ_{31}) ⊂ Ũ. It then follows that γ_{12}, γ_{23}, γ_{31} is a standard triangle in Ũ and Int_{U}(γ_{12}, γ_{23}, γ_{31}) = IntŨ(γ_{12}, γ_{23}, γ_{31}). Therefore the following is clear.

Lemma 4.3. Let (p_{1}, p_{2}, p_{3}, U) and (p_{1}, p_{2}, p_{3}, Ũ) be adjacent standard triples in M^{2}. Then σ(p_{1}, p_{2}, p_{3}, U) = σ(p_{1}, p_{2}, p_{3}, Ũ). (See Figure 4.) □

Note that for two standard triples to be adjacent all that matters is that they have a common standard triangle. They do not have to be close in any other sense, such as in the Hausdorff metric on closed sets.

p_{3}  p_{2} p^{1} γ_{12} γ_{31} γ_{23}

Figure 3. A standard triangle for (p_{1}, p_{2}, p_{3}, U) bounds a region of U. The orientation of M^{2} induces a boundary orientation on the standard triangle, which in turn induces a permutation σ(p_{1}, p_{2}, p_{3}, U) on the set {p_{1}, p_{2}, p_{3}}. As any two standard triangles are homotopic this permutation is independent of the choice of the standard triangle. (For the orientation shown the permutation is (p_{1}p_{2}p_{3}).)

5. Proof of the Theorem

Let M^{2} be a compact oriented surface smoothly immersed in R^{3} so that all the shadow sets S_{e} are simply connected. Then by Corollary 2.2 M^{2} is diffeomorphic to a sphere. Now assume, toward a contradiction, that M^{2} is not the boundary of a convex set. Then by Propositions 3.1 and 3.3 there is an open set V ⊂ S^{2} so that for every u ∈ V the height function h_{u} is a Morse function that has at least two local maximums. For each u ∈ V let u^{⊥} ∩ S^{2} be the great circle of S^{2} orthogonal to u. Then as u varies over V the circles u^{⊥} ∩ S^{2} will cover a set F of positive measure in S^{2}. By Proposition 2.1 there is an e ∈ F so that ∂S_{e} is a smooth curve in M^{2}. Now chose u ∈ V so that e ∈ u^{⊥} ∩ S^{2}. Then we have a pair u, e so that the height function

p_{3}  p_{2} p_{1} γ_{12} γ_{31} γ_{23}

Figure 4. Pictured are two adjacent standard triples (p_{1}, p_{2}, p_{3}, U) and (p_{1}, p_{2}, p_{3}, Ũ) with a common standard triangle. As σ(p_{1}, p_{2}, p_{3}, U) and σ(p_{1}, p_{2}, p_{3}, Ũ) are defined in terms of the standard triangle the equality σ(p_{1}, p_{2}, p_{3}, U) = σ(p_{1}, p_{2}, p_{3}, Ũ) holds.

h_{u} is a height function with two (or more) local maximums and an e⊥u so that ∂S_{e} is a smooth curve.

By rotation we can assume that u is in the direction of the positive z axis and e in the direction of the positive x axis. The height function h_{u} is then just the restriction of the z coordinate to M^{2}. Let e(θ) := (cos θ, sin θ, 0). Then e(0) = e. Let p_{1} and p_{2} be two local maxima of h_{u} and let p_{3} be a local minima. As u⊥e(θ) for all θ we see that p_{1}, p_{2}, p_{3} ∈ S_{e(θ)} for all θ.

By Proposition 3.1 the Gauss curvature of M^{2} is non-zero at all of the critical points p_{1}, p_{2} and p_{3} and thus the Gauss map g of M^{2} is a local diffeomorphism near each of these points. Now as p_{1} is a local maximum of h_{u} = z|_{M^{2}} the tangent plane to M^{2} at p_{1} is defined by z = c_{1} for some constant c_{1}. If δ > 0 is sufficiently small there is a connected component of M^{2} ∩ {z = c_{1} − δ} which is a simple closed curve on M^{2} that bounds a disk containing p_{1}. As the Gauss curvature is not zero at p_{1} the Gauss map g is a local diffeomorphism near p_{1}. By making δ small enough we can assume that g|_{D_{1}} : D_{1} → S^{2} is a diffeomorphism onto its image g[D_{1}]. For each θ we have that the shadow boundary ∂S_{e(θ)} is the preimage of the great circle e(θ)^{⊥}S^{2} under the Gauss map. As g|_{D_{1}} is a diffeomorphism this implies that D_{1} ∩ ∂S_{e(θ)} is a smooth curve through the point p_{1}. (This is pictured in Figure 5.) Letting c_{i} = h_{u}(p_{i}), we can construct similar disks D_{2} about p_{1} (with boundary a component of M^{2} ∩ {z = c_{2} − δ}) and D_{3} about p_{3} (with boundary a component of M^{2} ∩ {z = c_{3} + δ}) so that g|_{D_{i}}: D_{i} → S^{2} is a diffeomorphism for i = 1, 2, 3.

p_{1}  ∂S_{u(θ)} {x ∈ M^{2} : h^{u}(x) = c^{1} − δ}

Figure 5. Near the local maxima p_{1} a plane parallel to the tangent T (M)_{p_{1}} and a little below p_{1} will cut off a disk D_{1} in M^{2} so that the restriction of the Gauss map to D_{1} is a diffeomorphism with g[D_{1}] ⊂ S^{2}. Any of the shadow boundaries ∂S_{e(θ)} will meet U_{1} in a smooth curve.

As D_{i} ∩ ∂S_{e(θ)} is smooth for all θ and we are assuming that each S_{e(θ)} is simple connected we see that (p_{1}, p_{2}, p_{3}, S_{e(θ)}) is a nicely placed triple in the sense of Section 4.

Lemma 5.1. For any θ_{0} there is a ϵ > 0 so that if |θ − θ_{0}| < ϵ then (p_{1}, p_{2}, p_{3}, S_{e(θ)}) and (p_{1}, p_{2}, p_{3}, S_{e(θ_{0})}) are adjacent nicely placed triples.

Proof. We choose a standard triangle γ_{12}, γ_{23}, γ_{31} for S_{e(θ_{0})} so that at all the corners p_{1}, p_{2}, p_{3} the sides of the triangle form non-zero angles with ∂S_{θ_{0}}. In other terminology this is the same as requiring that all of the sides γ_{ij} are transverse to ∂S_{θ_{0}} (i.e. not tangent to ∂S_{θ_{0}}). (This is pictured in Figure 6.) By continuity if |θ_{0} − θ| is sufficiently small then ∂S_{e(θ)} will also be transverse to the sides γ_{ij}. Let T := \overline{Int(γ_{12}, γ_{23}, γ_{31})} ∩ S_{θ_{0}}. (Here \overline{A} is the closure of A in M^{2}.) From the definition of a standard triangle we have

T = \overline{Int(γ_{12}, γ_{23}, γ_{31})} \ \{p_{1}, p_{2}, p_{3}\}.

This discussion implies that there is an ϵ_{1} > 0 so that

(5.1)  T ∩ (D_{1} ∪ D_{2} ∪ D_{3}) ⊂ S_{θ}.

holds whenever |θ − θ_{0}| < ϵ_{1}.

∂S_{e(θ)} p_{1} D_{1} γ_{31}  γ_{12}  ∂S_{e(θ0)}

Figure 6. If γ_{12}, γ_{23}, γ_{31} is a standard triangle for (p_{1}, p_{2}, p_{3}, S_{e(θ_{0})}) so that at the corners p_{i} the sides γjk make a positive angle with ∂S_{e(θ_{0})} then the same will be true for ∂S_{e(θ)} with |θ − θ_{0}| sufficiently small. Therefore T ∩ D_{1} ⊂ S_{θ} for these θ.

The set E := T \ (D_{1} ∪ D_{2} ∪ D_{3}) is compact and f_{e(θ_{0})} is positive on E. Thus by compactness and continuity there is a ϵ_{2} > 0 so that f_{e(θ)} is positive on E for |θ − θ_{0}| < ϵ_{2}. But from the definition of S_{θ} in terms of f_{e(θ)} this implies

(5.2)  T \ (D_{1} ∪ D_{2} ∪ D_{3}) ⊂ S_{e(θ)}

for |θ − θ_{0}| < ϵ_{2}. Let ϵ = min{ϵ_{1}, ϵ_{2}}. Then (5.1) and (5.2) imply that T ⊂ S_{e(θ)} when |θ − θ_{0}| < ϵ. But this is just the condition that is required to show that S_{e(θ)} is adjacent to S_{e(θ_{0})}. □

Recall from Section 4 that for each θ the nicely placed triple (p_{1}, p_{2}, p_{3}, S_{e(θ)}) determines a permutation σ(p_{1}, p_{2}, p_{3}, S_{e(θ)}) of the set {p_{1}, p_{2}, p_{3}}. By possibly renaming p_{1} and p_{2} (the two local maxima) we can assume that when θ = 0 that σ(p_{1}, p_{2}, p_{3}, S_{e(0)}) = (p_{1}p_{2}p_{3}). Lemmas 4.3 and 5.1 imply that θ ↦ σ(p_{1}, p_{2}, p_{3}, S_{e(0)}) is locally constant. As the real numbers are connected this implies that σ(p_{1}, p_{2}, p_{3}, S_{e(θ)}) = (p_{1}p_{2}p_{3}) for all θ. Recall that for θ = 0 we have that S_{e(0)} has a smooth boundary ∂S_{e(0)}. By Lemma 4.2 the permutation σ(p_{1}, p_{2}, p_{3}, S_{e(0)}) is just the cyclic permutation induced on {p_{1}, p_{2}, p_{3}} by the boundary orientation of ∂S_{e(0)}. But e(π) = −e(0) and S_{e(π)} = M^{2} \ \overline{S}_{e(0)}. Therefore as sets ∂S_{e(0)} = ∂S_{e(π)}, but as the inward normal determined by S_{e(0)} is the outward normal determined by S_{e(π)} the boundary orientation ∂S_{e(0)} determined by S_{e(π)} is opposite of the one determined by S_{e(0)}. Therefore if σ(p_{1}, p_{2}, p_{3}, S_{e(0)}) = (p_{1}p_{2}p_{3}) then σ(p_{1}, p_{2}, p_{3}, S_{e(π)}) = (p_{1}p_{3}p_{2}) which contradicts that σ(p_{1}, p_{2}, p_{3}, S_{e(θ)}) = (p_{1}p_{2}p_{3}) for all θ. This completes the proof.

6. Tubes about Curves and an Example of a Genus One Surface with Connected Shadows

Let c: S^{1} → R^{3} be a smooth immersed curve. Let M^{2} be the tube of radius r about c. Then for r sufficiently small M^{2} is a smooth immersed torus in R^{3} (and if c is embedded M^{2} will be embedded). Explicitly

M^{2} := {c(t) + ru : t ∈ S^{1}, u ∈ S^{2}, u⊥c′(t)}.

The fiber of M^{2} over t is C_{t} := {c(t) + ru : u ∈ S^{2}, u⊥c′(t)} which is a circle. The Gauss map of M^{2} is given by

n(c(t) + ru) = u.

Therefore for any e ∈ S^{2} and t ∈ S^{1} the fiber C_{t} either is half inside (open

Figure 7. A tube about a curve that has no two parallel tangent lines. Such a tube has all its shadow sets connected.

half circle) and half outside (closed half circle) the shadow set S_{e} or C_{t} is entirely contained in the boundary of S_{e}. The latter only happens at points where c′(t)⊥e. Thus for each t the set C_{t} ∩S_{e} is either empty (when c′(t)⊥e) or C_{t} ∩ S_{e} is an open half circle. If [t_{0}, t_{1}] is an interval in S^{1} so that for no point t ∈ [t_{0}, t_{1}] does c′(t)⊥e hold, then it is easily seen that that part of the shadow S_{e} over [t_{0}, t_{1}] is connected. (That is S_{e} ∩ ⋃_{t∈[t_{0},t_{1}]} C_{t} is connected.) This implies that the number of connected components of S_{e} is at most the number of connected components of S^{1} \ {t ∈ S^{1} : c′(t)⊥e}. As at least two points of S^{1} must be removed before it is disconnected, if some shadow S_{e} is not connected there must be two values of t ∈ S^{1} with c′(t)⊥e.

Thus if the curve c has the property that its unit tangent curve t(t) := c′(t)/∥c′(t)∥ is a simple closed curve on the sphere that does not contain any of its antipodal points (or what is the same thing c has no two parallel tangent lines), then for each e ∈ S^{2} there is at most one t ∈ S^{1} so that c′(t)⊥e. Then the tube M^{2} will have all its shadow sets S_{e} connected. Figure 7 shows such a tube.

Figure 8. A small tube about the tangent image. This curve has the origin in the interior of its convex hull and also is disjoint from its antipodal image (refection through the origin).

As it is not obvious that a curve with these properties exists we now give a construction for one. Start with any simple closed curve t: S^{1} → S^{2} that does not contain any of its antipodal points and so that the origin is in the interior of the convex hull of t[S^{1}]. Figure 8 shows such a curve. This curve was constructed along the following lines. Start with a circle on the sphere which is parallel to a great circle. For instance, take a circle of constant latitude in the northern hemisphere, such as the Arctic circle, then construct three tongs (legs) which go down from the Arctic circle along three longitudes which are 120 degrees apart. Send these tongs below the equator, but stop just short of the Antarctic circle (the antipodal image of the Arctic circle).

The curve in Figure 8 has a parameterization

t(t) := [\begin{matrix}x(t) \\ y(t) \\ z(t) \end{matrix}] = [\begin{matrix}sin r(t) cos t \\ sin r(t) sin t \\ cos r(t) \end{matrix}]

where

r(t) := (.1 + .78 cos^{20}(3t/2))π.

(This defines the functions x(t), y(t), and z(t).) Then t is periodic of period 2π and thus can be viewed as defined on S^{1} = R/2πZ. But if a curve on the sphere has the origin in the interior of its convex hull then then it is the unit tangent curve of a closed space curve (see Propostion 7.1 for the exact statement and its proof). If c is a curve that has t as its unit tangent curve then c will have no parallel tangents and therefore a small tube about c will give an example of an embedded genus one surface in R^{3} with all shadow sets connected.

Figure 9. A top view of the tube of Figure 7 showing that its central curve lies on a cylinder over a convex plane curve.

The proof of Proposition 7.1 is contructive and we can use it to explicitly find the curve c. First let

v(t) := .1 + a cos^{20}(3t/2)

where a is a positive number to be chosen shortly. Then for symmetry reasons (or direct calculation)

∫_{0}^{2π} v(t)x(t) dt = ∫_{0}^{2π} v(t)y(t) dt = 0.

Now choose a so that

∫_{0}^{2π} v(t)z(t) dt = 0.

Then a ≈ .8824923325 … to enough accuracy to produce decent graphics. The integrals above and the periodicity of t imply that if

c(t) := ∫_{0}^{t} v(τ)t(τ) dτ

then c(t + 2π) = c(t). Therefore c: S^{1} → R^{3} is a closed curve and c′(t) = v(t)t(t) and so t is the unit tangent curve of t. Then, as t does not meet its antipodal curve, c will not have any two parallel tangent lines and gives us our required curve. Figure 7 is the tube of radius r = .01 about this curve.

We leave it as an exercise to show that the projection of c onto the xy plane is a convex curve. This can be seen in Figure 9 which is the tube of Figure 7 viewed from high above the x-y plane with the direction of sight parallel to the z axis.

7. Appendix: A Sufficient Condition for a Spherical Curve to be the Tangent Mapping of a Space Curve

The following sufficient condition for a closed curve on a sphere to be the tangent map to a space curve is, in the proper circles, a well known folk theorem (I learned of the result and its proof from Mike Gage), however the only explicit reference I have seen is the original paper of Fenchel on the total curvature of space curves (Uber Krummungund Windung geschlossener Raumkurven, Math. Ann. 101 (1929) p. 238–252) where it is stated without proof and attributed to Löwner^{1}.

Proposition 7.1. Let t: S^{1} → S^{2} be a smooth map so that the origin is in the interior of the convex hull of the image of t. Then there is an immersed space curve c: S^{1} → R^{3} that has t as unit tangent map (that is t(t) = c′(t)/∥c′(t)∥).

Remark 7.2. It is all right if t is not injective, or if it is not an immersion.

Proof. We view S^{1} as R/Z so that t can be thought of as a map t: R → S^{2} with t(t + 1) = t(t). If we can find a positive function v: R → (0, ∞) with v(t + 1) = v(t) and so that the “center of gravity” of the product vt is the origin, that is

(7.1)  ∫_{0}^{1} v(τ)t(τ) dτ = 0.

1Fenchel attributes the result to Löwner and gives as reference Pólya and Szegő Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie, Springer, Berlin, (1925) S. 165 und 391 Aufgabe, 13. But in fact Pólya and Szegő only show that the origin is in the convex hull of the image of a tangent mapping and do not state or prove the converse. They credit this result to Löwner but without explicit reference.

Then periodicity implies that for all t

(7.2)  ∫_{t}^{t+1} v(τ)t(τ) dτ = 0.

Set

c(t) := ∫_{0}^{t} v(τ)t(τ) dτ.

The integral condition (7.2) implies c(t + 1) = c(t) so c is a closed curve c: S^{1} → R^{3}. Also c′(t) = v(t)t(t) and as v is positive this implies that c is an immersion. Finally c′(t) = v(t)t(t) clearly implies that t is the unit tangent map to c.

We now show the existence of the function v so that (7.1) holds. As the origin is in the interior of the convex hull of c there are distinct t_{0}, t_{1}, t_{2}, t_{3} ∈ [0, 1) and α_{0}, α_{1}, α_{2}, α_{3} ∈ (0, 1) with each α_{i} positive, ∑_{i=0}^{3} α_{i} = 1 and

∑_{i=0}^{3} α_{i}t(t_{i}) = 0.

By continuity there is a δ > 0 so that if P_{0}, P_{1}, P_{2}, P_{3} ∈ R^{3} then

(7.3) ∥t(t_{i}) − P_{i}∥ < δ for i = 1, …, 4 ⟹ 0 ∈ convex hull of P_{0}, …, P_{3}.

Now for i = 0, …, 4 there is a smooth positive C^{∞} function v_{i} on R with v_{i}(t + 1) = v_{i}(t), ∫_{0}^{1} v_{i}(t) dt = 1 and that approximates the point mass at t_{i} well enough that

∥t(t_{i}) − ∫_{0}^{1} v_{i}(t)t(t) dt∥ < δ.

Therefore the implication (7.3) yields β_{0}, …, β_{3} ≥ 0 with ∑_{i=0}^{3} β_{i} = 1 and

0 = ∑_{i=0}^{3} β_{i} ∫_{0}^{1} v_{i}(t)t(t) dt = ∫_{0}^{1} ∑_{i=0}^{3} β_{i}v_{i}(t)t(t) dt = ∫_{0}^{1} v(t)t(t) dt

where v(t) = ∑_{i=0}^{3} β_{i}v_{i}(t). This gives us the v so that the desired relation (7.1) holds and completes the proof. □

Acknowledgment: I would like to thank Sara Kasiri for suggesting some improvements to these notes.

References

1. S. S. Chern and R. L. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306–318.

2. , On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5–12.

3. J. Milnor, Morse theory, Princeton University Press, Princeton, N.J., 1963, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51.