Let X: M^{n} -> R^{n+1} be an immersion. For each unit vector V in R^{n+1}, let g_{V} be the hyperplane through the origin perpendicular to V, and let ^{V}: M^{n} -> g_{V} be the composition of X with orthogonal projection to g_{V}:
^{V}(P) = X(P) - (X(P) ^{.} V) V
Let TS^{n} be the tangent bundle of S^{n}. Identifying g_{V} with the tangent hyperplane to S^{n} at V, we obtain a family of mappings parametrized by S^{n}:^{*}:S^{n} x M^{n} -> TS^{n}, ^{*}(V, P) = (V, ^{V}(P))
The critical set C of the family ^{*} is the set of pairs (V, P) such that V is parallel to the tangent hyperplane to X at P, so C is identified with the unit tangent sphere bundle of X.Since the tangent bundle of S^{n} is trivial over the complement of any point V_{0} S^{n}, the restriction of ^{*} to (S^{n} - {V_{0}}) x M^{n} is an unfolding of ^{V} for all V V_{0}. 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: M^{n} -> R^{n+1} with the orthogonal projection R^{n+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: R^{n} -> R^{n} of kernel rank at most one, S^{1}(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^{1k}(f) is the set of singular points of f|_{S1k-1(f)}. Also S^{1k,0}(f) denotes S^{1k}(f) - S^{1k-1}(f). Thus S^{1,0}(f) is the fold locus of f, where the kernel field of Df is not tangent to S^{1}(f). The locus S^{1,1}(f) where is tangent to S^{1}(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^{1k}(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:R^{n} -> R^{n} 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^{1k}(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^{1k+1}(f). To see the connection with kernel vector fields, consider the equations for S^{1,1}(f). They are
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 S^{1}(f)) this kernel field is tangent to S^{1}(f).
If f(x_{1}, ..., x_{n}) = (x_{1}, ..., x_{n-1}, f^{n}(x_{1}, ..., x_{n})), with grad f^{n} 0, the equations defining S^{1k} become much simpler. Then P S^{1k,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^{1k, 0}(^{V}):
A point P M^{n} is in S^{1k, 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 R^{n+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 R^{n+1} so that X(x_{1}, ..., x_{n}) = (x_{1}, ..., x_{n}, f(x_{1}, ..., f_{n})), and l is spanned by (0, ..., 0, 1, 0). Then
which has dimension k+1, where (0) = 0, ik, (0) = 0. But this is the same condition given above for P S^{1k, 0}(^{V}).
Now we examine the family ^{*} for X: M^{2} -> R^{3} 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: M^{2} -> R^{3}, then a line in R^{3} 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 R^{3} 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 R^{3} we may assume that there is a coordinate neighborhood about P on which X has the form
X(x, y) = (x, y, x^{2}/2 + g(x, y)), P = (0, 0),
where the germ of g at zero is in (_{2})^{3}. Here _{2} is the maximal ideal of the local ring _{0}(R^{2}) of germs at zero of real-valued functions on R^{2}. The constant is the nonzero principal curvature of X at P, and (x, y, x^{2}) is the osculating paraboloid of X at P. The principal direction associated to is the x-axis, and the zero principal curvature direction is the y-axis. If P is a cusp of the Gauss map N, then P S^{1, 1}(N), so g_{yyy}(0) = 0, by a computation using the modified Gauss map Ñ (Chapter 1) and the equations (i) (ii) above for S^{1,1}(N). But g_{yyy}(0) = 0 implies that the y-axis has order of contact 3 with X at 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 ^{*}: S^{2} x M^{2} -> TS^{2} 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
Consider the orthogonal projection to a plane containing the z-axis:^{V}(x, y) = (-bx + ay, f(x, y))
where V = (a, b) is a unit tangent vector to X at the origin.Let D^{n}f: (R^{2})^{n} -> R be the symmetric multilinear function whose coefficients are the mixed partial of f of order n.
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), D^{2}f(V, V), and D^{2}f(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 D^{2}f(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, D^{2}f(V, V)). (It is not necessary to use D^{2}f(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 D^{3}f(V, V, V). So 0 S^{13}(^{V}) if and only if 0 S^{12}(^{V}) and D^{3}f(V, V, V).
(iii) has a similar proof.
Now we look at the relation between S^{1k}(^{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 .
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
D^{2}f((x))['(s), '(s)] = 0
so
0 = (D^{2}f((s))['(s), '(s)])'
= D^{3}f((s))['(s), '(s), '(s)] + 2 D^{2}f((s))['(s), ''(s)]
= D^{3}f((s))['(s), '(s), '(s)] + 2 _{}(s)D^{2}f((s))['(s),
n(s)]
where n is the unit normal vector of . Now 0 S^{12,0}(^{V}) if and only if D^{3}f(0)(V, V, V) 0, so 0 S^{12,0}(^{V}) if and only if 2_{}(0) D^{2}f(0)['(0), n(0)]0, so _{}(0)0. Note that D^{2}f(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) D^{3}f((0))['(0), '(0), '(0)] = 0 iff _{(0)} = 0 iff 0 S^{13}(^{V}). Now
0 = (D^{2}f((s))['(s), '(s)])''
= D^{4}f()[', ', ',
'] + 3D^{3}f()[', ',
n] + 2D^{3}f()[', ',
n]
+ 2^{2}D^{2}f()[n, n] +
2D^{2}f()[', n' + 'n]
If s = 0 and 0 S^{13}(^{V}), then
0 = D^{4}f(0)[V, V, V, V] + 2D^{2}f(0)[V, '(0), n(0)]
So 0 S^{13,0}(^{V}) iff '(0) 0.(iii) Compute 0 = (D^{2}f((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:
In geometric terms, if l is the line through P parallel to V, we have:
We can now prove half of theorem 3.1(h).
Theorem 7.5 Let X: M^{2} -> R^{3} 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 M^{2} be a smooth surface. For an open dense subset C of the space of immersions X: M^{2} -> R^{3}, the germ of the family ^{*} at (V, P) is a versal unfolding of the germ of ^{V} at P for all (V, P) S^{2} x M^{2}.
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, x^{2} y - x^{2}) 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 R^{3} 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 R^{3} 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 S^{2} x M^{2}, 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 M^{2} 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}(^{*}) -> S^{2}, the restriction of the projection S^{2} x M^{2} -> S^{2}.
Proof See [GafR].
Theorem 7.10 If X: M^{2} -> R^{3} 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}: S^{1}(^{*}) -> S^{2} be the "fold catastrophe map" of the family ^{*}, i.e. the restriction of the projection S^{2} x M^{2} -> S^{2}. 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) S^{1}(^{*}). 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 R^{2} in R^{3}. 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: R^{2} -> R^{3} 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^{k}X(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) =1/2x^{2} + a_{04}y^{4} + a_{12}xy^{2} + a_{21}x^{2}y + 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 a_{12}^{2} - 2a_{04} 0, a_{12} 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
Now the curvature of the parabolic image curve of X is determined by j^{2}(X°), where is a parametrization of the parabolic curve. This 2-jet is determined by the 2-jets of X and . Applying the chain rule and standard formulas for the curvature of a space curve we obtain that the curvature of the parabolic image curve at 0, for an immersion X such that N has a cusp at 0, is
Thus is j^{k}X(0) V, then the curvature of the parabolic image curve of X at 0 is nonzero if and only if (a_{12})^{2} - 3a_{04} 0
By [BlW] and lemma 7.12, there exists an algebraic set Y of codimension 3 in Imm^{4}(2, 3) such that j^{4}X(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: M^{2} -> R^{3},
is easily obtained by considering the space of 4-jets of immersion
Imm^{4}(M^{2}, R^{3}) as a
fiber bundle over M, with fiber the space of 4-jets at zero
of immersions of R^{2} in R^{3}.