Chapter 7

Projections to planes

Let X: Mⁿ -> Rⁿ⁺¹ be an immersion. For each unit vector V in Rⁿ⁺¹, let g_V be the hyperplane through the origin perpendicular to V, and let ^V: Mⁿ -> g_V be the composition of X with orthogonal projection to g_V:

^V(P) = X(P) - (X(P) ^. V) V

^*:Sⁿ  x  Mⁿ -> TSⁿ, ^*(V, P) = (V, ^V(P))

Since the tangent bundle of Sⁿ is trivial over the complement of any point V₀ Sⁿ, the restriction of ^* to (Sⁿ - {V₀}) x Mⁿ is an unfolding of ^V for all V V₀. Since ^V is a mapping of n-manifolds, its singularities can be more complicated than the singularities of a real-valued function (such as the height function g_V studied in chapter 5). Since ^V is the composition of the immersion X: Mⁿ -> Rⁿ⁺¹ with the orthogonal projection Rⁿ⁺¹ -> g_V, it follows that ^V has kernel rank at most one. But ^V may have cuspoid (Morin) singularities of arbitrarily high order. To get a geometric interpretation of these singularities, we first review their definition.

Thom [T1] suggested that for a "generic" mapping f: N -> P one could stratify the source N by the kernel rank of f. Each resulting stratum S would in turn be stratified by the kernel rank of f|_S. This process would be repeated until no new strata were produced. For a generic map f: Rⁿ -> Rⁿ of kernel rank at most one, S¹(f) denotes the set of singular points of f, S^1,1(f) = S^1₂(f) denotes the set of singular points of f|_S¹(f), and in general S^1_k(f) is the set of singular points of f|_S^1k-1(f). Also S^1_k,0(f) denotes S^1_k(f) - S^1_k-1(f). Thus S^1,0(f) is the fold locus of f, where the kernel field of Df is not tangent to S¹(f). The locus S^1,1(f) where is tangent to S¹(f) is then subdivided into the cusp points S^1,1,0(f) where is not tangent to S^1,1(f), and the points S^1,1,1(f) where is tangent to S^1,1(f), and so on. The stratum S^1₃,0 is the swallowtail points of f, and S^1₄,0(f) is the butterfly points of f. The dimension of S^1_k(f) is n - k.

Boardman carried out Thom's suggestion rigorously. He defined submanifolds of the infinite jet space J(N, P) whose pullbacks by the jet extension map of f are the desired strata of n. A "generic" map of f is one whose jet extension is transverse to these Boardman submanifolds. It is possible to write down local defining equations for these submanifolds, and so it is possible to stratify nongeneric maps as well. For mappings f:Rⁿ -> Rⁿ of corank 1, it is easy to give an inductive construction of these equations (cf. [Mori] and [Mat] for details). Start with the equations of S^1_k(f) and the component functions of one map F. Then the n x n minors of the Jacobian matrix DF are the equations of S^1_k+1(f). To see the connection with kernel vector fields, consider the equations for S^1,1(f). They are

(i)

det Df = 0

(ii)

(grad det Df) ^. (()', ..., ()') = 0, 1r n

where ()' denotes in Df. For a mapping f of rank n-1, one of the vectors (()', ...,()') must span the kernel of Df. Condition (ii) implies that for generic f (i.e. grad det Df 0 on S¹(f)) this kernel field is tangent to S¹(f).

If f(x₁, ..., x_n) = (x₁, ..., x_n-1, fⁿ(x₁, ..., x_n)), with grad fⁿ 0, the equations defining S^1_k become much simpler. Then P S^1_k,0(f) if and only if (P) = 0 for 1 i k and (P) 0. This allows a direct geometric interpretation of the Thom-Boardman strata S^{1_k, 0}(^V):

A point P Mⁿ is in S^{1_k, 0}(^V) if and only if the line through X(P) parallel to V has k^th order contact with the immersion X at P.

Since any immersed hypersurface is locally a function graph, this statement follows from [St, p. 24, (7-4)]. Or we can define the order of contact of a hypersurface and a line to be m-1, where m is their intersection multiplicity. This multiplicity m is the dimension of the real vector space _A/I, where _A is the local ring of germs at A = X(P) of real-valued functions on Rⁿ⁺¹, and I is the ideal generated by the local defining equations of X(M) and the line. If this line l is not tangent ot X(M) at A, then m is one. If it is tangent, then there are coordinate charts about P in M and X(P) in Rⁿ⁺¹ so that X(x₁, ..., x_n) = (x₁, ..., x_n, f(x₁, ..., f_n)), and l is spanned by (0, ..., 0, 1, 0). Then

Now we examine the family ^* for X: M² -> R³ in more detail. The following result implies theorem 3.1(g).

Theorem 7.1 If P is a cusp of the Gauss map of the immersion X: M² -> R³, then a line in R³ has order of contact > 2 with X at P. Conversely, if X A, and P is a parabolic point of X, and there exists a line in R³ which has order of contact >2 with X at P, then P is a cusp of the Gauss map of X.

Proof If P is a parabolic point of X, then after a rigid motion of R³ we may assume that there is a coordinate neighborhood about P on which X has the form

X(x, y) = (x, y, x²/2 + g(x, y)),   P = (0, 0),

Conversely, if a line has order of contact 2 with X at 0, it must also have order of contact 2 with the osculating paraboloid so it must be the y-axis. If the y-axis has order of contact 3 with X at 0 then g_yyy(0) = 0, so P S^1,1(N). Thus if X A then P is a cusp of N.

In order to relate cusps of Gauss mappings to the geometry of asymptotic curves, we first investigate the relation between the singularities of the map ^*: S² x M² -> TS² and the second fundamental form of X.

We assume that the immersion X is locally of the form

X(x, y) = (x, y, f(x, y)), grad f(0, 0) = 0

^V(x, y) = (-bx + ay, f(x, y))

Let Dⁿf: (R²)ⁿ -> R be the symmetric multilinear function whose coefficients are the mixed partial of f of order n.

Proposition 7.2

(i)

0 S^1₂(^V) iff V is an asymptotic vector of X at 0.

(ii)

0 S^1₃(^V) iff D³f(V, V, V) = 0 and 0 S^1₂(^V).

(iii)

0 S^1₄(^V) iff D⁴f(V, V, V, V) = 0 and 0 S^1₃(^V).

Proof (i) Let F(x, y) = (-bx + ay, f(x, y), det D^V(x, y)). The equations which define S^1₂(^V) are the 2 x 2 minors of DF. These minors are Df(V), D²f(V, V), and D²f(V, (-f_y, f_x)). The first and third of these minors are zero at 0 so only the second is a new condition. But D²f(V, V) = 0 if and only if the second fundamental form II(V, V) = 0, i.e. V is an asymptotic vector.

(ii) The defining equations of S^1₃ are the 2 x 2 minors of DG, where G has component functions (F, D²f(V, V)). (It is not necessary to use D²f(V, (-f_y, f_x)) since this function is in the ideal generated by the components of G.) These minors are in the ideal generated by the minors of DF, and D³f(V, V, V). So 0 S^1₃(^V) if and only if 0 S^1₂(^V) and D³f(V, V, V).

(iii) has a similar proof.

Now we look at the relation between S^1_k(^V) and asymptotic curves. Recall that X(x, y) = (x, y, f(x, y)), grad f(0, 0) = 0. Assume that (0, 0) is a hyperbolic point of X, and is an asymptotic curve of X in the (x, y) plane, parametrized by arc-length, with '(0) = V. Let (s) be the curvature of .

Proposition 7.3

(i)

0 S^{1₂, 0}(^V) iff (0) 0.

(ii)

0 S^{1₃, 0}(^V) iff (0) = 0, '(0) 0.

(iii)

0 S^{1₄, 0}(^V) iff (0) = 0, '(0) = 0, ''(0) 0.

Proof We proceed by the time-honored principle of differentiating something which is identically zero.

(i) is an asymptotic curve of X if and only if

D²f((x))['(s), '(s)] = 0

so

0 = (D²f((s))['(s), '(s)])'

= D³f((s))['(s), '(s), '(s)] + 2 D²f((s))['(s), ''(s)]

= D³f((s))['(s), '(s), '(s)] + 2 (s)D²f((s))['(s), n(s)]

where n is the unit normal vector of . Now 0 S^1₂,0(^V) if and only if D³f(0)(V, V, V) 0, so 0 S^1₂,0(^V) if and only if 2(0) D²f(0)['(0), n(0)]0, so (0)0. Note that D²f(0)['(0), n(0)] 0 since '(0) and n(0) are orthogonal, for the only orthogonal conjugate directions at a hyperbolic point are the principal directions. This proves (i).

(ii) D³f((0))['(0), '(0), '(0)] = 0 iff ₍₀₎ = 0 iff 0 S^1₃(^V). Now

0 = (D²f((s))['(s), '(s)])''

= D⁴f()[', ', ', '] + 3D³f()[', ', n] + 2D³f()[', ', n]

+ 2²D²f()[n, n] + 2D²f()[', n' + 'n]

If s = 0 and 0 S^1₃(^V), then

0 = D⁴f(0)[V, V, V, V] + 2D²f(0)[V, '(0), n(0)]

(iii) Compute 0 = (D²f((s))['(s), '(s)])'''.

Now consider the space curve X °  with curvature _X ° . Since an asymptotic curve on a surface has normal curvature zero, its curvature as a space curve equals its intrinsic curvature. By our choice of coordinates for the immersion X, _X ° (0) = (0) provided that (0) for j<i. Thus we have:

Corollary 7.4

(i)

0 S^{1₂, 0}(^V) iff _X ° (0) 0.

(ii)

0 S^{1₃, 0}(^V) iff _X ° (0) = 0, '_X ° (0) 0.

(iii)

0 S^{1₄, 0}(^V) iff _X ° (0) = 0, '_X ° (0) = 0, ''_X ° (0) 0.

In geometric terms, if l is the line through P parallel to V, we have:

(i)

l has order of contact 2 with X at P iff P S^1₂,0(^V) iff V is an asymptotic vector of X at P, and the corresponding asymptotic curve X °  has nonzero intrinsic curvature at P.

(ii)

l has order of contact 3 with X at P iff P S^1₃,0(^V) iff V is an asymptotic vector of X at P, and the corresponding asymptotic curve X °  has a simple inflection at P. (Such a line l is called "flecnodal" in [Sa, 588].

(iii)

l has order of contact 4 with X at P iff P S^1₄,0(^V) iff V is an asymptotic vector of X at P, and the corresponding asymptotic curve X °  has a simple "undulation" at P.

We can now prove half of theorem 3.1(h).

Theorem 7.5 Let X: M² -> R³ be an immersion, with X A. If P is a parabolic point of X which is in the closure of the set of asymptotic inflection points of X, then P is a cusp of the Gauss mapping of X.

Proof By corollary 7.4, P is in the closure of S^1₃,0(^V), which is S^1₃(^V). Therefore the line through X(P) parallel to V has order of contact > 3 with X at P. By theorem 7.1, P is a cusp of the Gauss mapping.

A more precise analysis of the singularities of the family ^* is based on the following result.

Theorem 7.6 (Arnold, Lyashko, Goryunov, Gaffney-Ruas) Let M² be a smooth surface. For an open dense subset C of the space of immersions X: M² -> R³, the germ of the family ^* at (V, P) is a versal unfolding of the germ of ^V at P for all (V, P) S² x M².

For proofs, see [GafR] [A6]. (This result is closely related to theorem (A) of [Wa2, p. 712]). If the germ of ^* at (V, P) is a versal unfolding of the germ of ^V at P, then the germ of the mapping ^* at (V, P) is stable. The converse is not true. If the germ of ^* is versal at each point, then the germ of the Gauss map of X is stable at each point, so C is a subset of A. Menn's surface X(x, y) = (x, y, x² y - x²) has a stable Gauss map at (0, 0), but ^* is not versal at (0, 0).

Gaffney and Ruas' proof is based on an explicit classification of all rank one finitely determined germs of codimension four or less. For X C, there are ten equivalence classes of germs which may occur as germs of ^V at P (see [GafR] [A6]). For example, if X C then S^1₅(^V) = Ø and S^1₄(^V) is a set of isolated points. Furthermore, P S^1₄(^V) only if P is hyperbolic. Therefore a line l in R³ can have order of contact at most 3 with X at a parabolic point, so theorem 7.1 can be strengthened for X C:

Corollary 7.7 If X C, then P is a Gaussian cusp if and only if P is a parabolic point of X and there is a line in R³ which has third order contact with X at P.

Corollary 7.8 If X C, then P is a Gaussian cusp if and only if P is a parabolic point of X in the closure of the set of inflection points of asymptotic curves.

Proof By the proof of theorem 7.1, if X C, then P is a cusp of the Gauss mapping if and only if (V, P) S^1₃(^*), where V is asymptotic at P. Since ^* is versal at (V, P) and (V, P) and S^1₃ is a codimension 3 singularity, (V, P) S^1₃(^*) if and only if there exists a curve (V(t), P(t)) in S² x M², 0 t , such that P(t) S^1₃,0(^V(t)) for t > 0, and (V(0), P(0)) = (V, P). The projection of this curve to M² is a curve of asymptotic inflection points with P in its closure.

Using theorem 7.6 we can also complete the proofs of theorem 3.1(h) and theorem 3.3. It will be necessary to use a few facts about the "cusp catastrophe map" ^1,1: S^1,1(^*) -> S², the restriction of the projection S² x M² -> S².

Proposition 7.9 (Gaffney and Ruas) Let X: M² -> R³ be an immersion, with M orientable. If X C then

(i)

S^1,1(^*) is diffeomorphic to a two-sheeted cover of the double of the manifold of negative curvature of M.

(ii)

The singularity set of ^1,1 is the union of S^1₃(^*) and the set {(V, P)|P is parabolic and V is an asymptotic vector at P }.

(iii)

If P is a cusp of the Gauss map and V is an asymptotic vector at P, then the germ of ^1,1 at (V, P) is equivalent to (x, y³ - x²y). (Note that this germ is not stable.)

Proof See [GafR].

Theorem 7.10 If X: M² -> R³ is an immersion, and P is a cusp of the Gauss mapping, then P is in the closure of the set of asymptotic inflection points of X.

Proof By hypothesis there exists a neighborhood U of P such that N|_U is stable. Assume that P is the only cusp of N|_U, and the double of the part of U with negative curvature is a disc. By theorem 7.6 there exists a sequence of immersions X_n such that X_n|_U converges to X|_U in the Whitney topology and the family _n^* associated to X_n is versal on U. Since N is stable on U, we can assume that, for n sufficiently large the Gauss map N_n of X_n differs from N by a coordinate change in the source and target. Thus for n sufficiently large, the double of the part of U on which X_n has negative curvature is a disc, so S^1,1(_n^*) is two discs. Let D_n be one of these discs. Let _n be the curve on D_n which consists of points of S^1,1,1(_n^*). Let _n be the curve in D_n lying over the parabolic curve S_n. The curve _n divides D_n into two discs, each of which projects diffeomorphically to the negatively curved part of U. By proposition 7.9, _n crosses _n transversely at a single point, which lies over the cusp of N_n.

The projection of _n to U is a curve _n which is smooth except perhaps at the cusp of N_n, and which consists of inflection points of asymptotic curves of X_n. Since _n crosses _n transversely at a single point lying over the cusp of N_n, the limit of _n as n goes to infinity must be an infinite set I containing the cusp of N in its closure.

Since the jets of ^* at the points of I are in the closure of the jets of S^1₃(_n^*), they must be in S^1₃(^*), and since N|_U is stable, the only parabolic point of I is the Gaussian cusp P, by theorem 6.5. By corollary 7.4, all the other points of I are asymptotic inflections.

This proof illustrates a useful technique: first prove a theorem under a stringent genericity condition; then relax this condition, and use the relaxed condition to control the degeneration of the geometry.

Our final characterization of the Gaussian cusps, theorem 3.3, reflects the relationship between the two "catastrophe maps" of the family ^*. Let ¹: S¹(^*) -> S² be the "fold catastrophe map" of the family ^*, i.e. the restriction of the projection S² x M² -> S². Thus we have a diagram

Proposition 7.11 (Gaffney and Ruas) If X C then the germ at (V, P) of ¹ is stable for all (V, P) S¹(^*). This germ is singular if and only if P is parabolic.

Proof see [GafR].

So if X C the bifurcation set of ¹ is the asymptotic image of the parabolic curve. By proposition 7.9, the bifurcation set of ^1,1 is the union of and the asymptotic image of the asymptotic inflection curve. By 7.9(iii) the intersection points of and are precisely the asymptotic images of the Gaussian cusps. In particular, if X C then the asymptotic image of the parabolic curve of X is regular at Gaussian cusps. This completes the proof of theorem 3.3.

Finally, we finish the proof of theorem 3.2. Let Imm^k(2, 3) be the space of k-jets of immersion of R² in R³. Since Imm^k(2, 3) is a Euclidean space, with coordinates the partial derivatives of order  k, we can consider algebraic subsets of Imm^k(2, 3), i.e. subsets defined by polynomial equations. Recall that the parabolic image curve of the immersion X: R² -> R³ is the restriction of X to the parabolic curve of X.

Lemma 7.12 If k 4, there exists an irreducible algebraic subset V of codimension 2 in Imm^k(2, 3), and a proper algebraic subset W of V, such that j^kX(0) V - W if and only if the Gauss mapping N of X is stable in some neighborhood of 0, N has a cusp at 0, and the curvature of the parabolic image curve of X at 0 is nonzero.

Proof It suffices to consider immersion of the form X(x, y) = (x, y, f(x, y)), f (₂)². The set V will be all k-jets of immersions X with f(x, y) =1/2x² + a₀₄y⁴ + a₁₂xy² + a₂₁x²y + g(x, y) where g (₂)³ and g_yyyy(0) = g_yyy(0) = g_xyy(0) = g_xxy(0) = 0 (cf. the proof of theorem 7.1). The condition that the Gauss map N be stable is a₁₂² - 2a₀₄ 0, a₁₂ 0, 0. (This can be verified by direction computation.) To obtain that the curvature of the parabolic image curve of X at 0 is nonzero, we have to throw away another proper algebraic subset of V.

By considering the determinant of the Hessian of f, it is possible to solve implicitly for the 2-jet of the preimage of the parabolic curve in the parametrization plane. It is

By [BlW] and lemma 7.12, there exists an algebraic set Y of codimension 3 in Imm⁴(2, 3) such that j⁴X(P) Y if and only if (i) N is stable in a neighborhood of P, and (ii) N has a cusp at P implies the curvature of the parabolic image curve of X at P is nonzero. So the Thom transversality theorem implies the last statement in theorem 3.2, completing the proof of this theorem.

A global version of theorem 3.2, for an immersion X: M² -> R³, is easily obtained by considering the space of 4-jets of immersion Imm⁴(M², R³) as a fiber bundle over M, with fiber the space of 4-jets at zero of immersions of R² in R³.