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# Chapter 7 Projections to planes

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

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

Let TSn be the tangent bundle of Sn. Identifying gV with the tangent hyperplane to Sn at V, we obtain a family of mappings parametrized by Sn:

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

The critical set C of the family * is the set of pairs (VP) 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 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 gV studied in chapter 5). Since V is the composition of the immersion XMn -> Rn+1 with the orthogonal projection Rn+1 -> gV, 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 fN -> 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 fRn -> Rn of kernel rank at most one, S1(f) denotes the set of singular points of f, S1,1(f) = S12(f) denotes the set of singular points of f|S1(f), and in general S1k(f) is the set of singular points of f|S1k-1(f). Also S1k,0(f) denotes S1k(f) - S1k-1(f). Thus S1,0(f) is the fold locus of f, where the kernel field of Df is not tangent to S1(f). The locus S1,1(f) where is tangent to S1(f) is then subdivided into the cusp points S1,1,0(f) where is not tangent to S1,1(f), and the points S1,1,1(f) where is tangent to S1,1(f), and so on. The stratum S13,0 is the swallowtail points of f, and S14,0(f) is the butterfly points of f. The dimension of S1k(f) is n - k.

Boardman carried out Thom's suggestion rigorously. He defined submanifolds of the infinite jet space J(NP) 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 S1k(f) and the component functions of one map F. Then the n x n minors of the Jacobian matrix DF are the equations of S1k+1(f). To see the connection with kernel vector fields, consider the equations for S1,1(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, ..., xn) = (x1, ..., xn-1, fn(x1, ..., xn)), with grad fn 0, the equations defining S1k become much simpler. Then P S1k,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 S1k, 0(V):

A point P Mn is in S1k, 0(V) if and only if the line through X(P) parallel to V has kth 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, ..., xn) = (x1, ..., xn, f(x1, ..., fn)), 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 S1k, 0(V).

Now we examine the family * for XM2 -> 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 XM2 -> 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(xy)),   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(R2) of germs at zero of real-valued functions on R2. The constant is the nonzero principal curvature of X at P, and (x, y, x2) 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 S1, 1(N), so gyyy(0) = 0, by a computation using the modified Gauss map Ñ (Chapter 1) and the equations (i) (ii) above for S1,1(N). But gyyy(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 gyyy(0) = 0, so P S1,1(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(xy) = (xyf(xy)), grad f(0, 0) = 0

Consider the orthogonal projection to a plane containing the z-axis:

V(xy) = (-bx + ay, f(x, y))

where V = (a, b) is a unit tangent vector to X at the origin.

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

(i)
0 S12(V) iff V is an asymptotic vector of X at 0.
(ii)
0 S13(V) iff D3f(VVV) = 0 and 0 S12(V).
(iii)
0 S14(V) iff D4f(VVVV) = 0 and 0 S13(V).

Proof (i) Let F(xy) = (-bx + ayf(xy), det DV(xy)). The equations which define S12(V) are the 2 x 2 minors of DF. These minors are Df(V), D2f(VV), and D2f(V, (-fy, fx)). 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(VV) = 0, i.e. V is an asymptotic vector.

(ii) The defining equations of S13 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, (-fy, fx)) 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 S13(V) if and only if 0 S12(V) and D3f(V, V, V).

(iii) has a similar proof.

Now we look at the relation between S1k(V) and asymptotic curves. Recall that X(xy) = (xyf(xy)), grad f(0, 0) = 0. Assume that (0, 0) is a hyperbolic point of X, and is an asymptotic curve of X in the (xy) plane, parametrized by arc-length, with '(0) = V. Let (s) be the curvature of .

(i)
0 S12, 0(V) iff (0) 0.
(ii)
0 S13, 0(V) iff (0) = 0, '(0) 0.
(iii)
0 S14, 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 S12,0(V) if and only if D3f(0)(V, V, V) 0, so 0 S12,0(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 S13(V). Now

0 = (D2f((s))['(s), '(s)])''

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

+ 22D2f()[n, n] + 2D2f()[', n' + 'n]

If s = 0 and 0 S13(V), then

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

So 0 S13,0(V) iff '(0) 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:

(i)
0 S12, 0(V) iff X ° (0) 0.
(ii)
0 S13, 0(V) iff X ° (0) = 0, 'X ° (0) 0.
(iii)
0 S14, 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 S12,0(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 S13,0(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 S14,0(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 XM2 -> 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 S13,0(V), which is S13(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 XM2 -> R3, the germ of the family * at (VP) is a versal unfolding of the germ of V at P for all (VP) 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 (VP) is a versal unfolding of the germ of V at P, then the germ of the mapping * at (VP) 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(xy) = (xyx2 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 S15(V) = Ø and S14(V) is a set of isolated points. Furthermore, P S14(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 (VP) S13(*), where V is asymptotic at P. Since * is versal at (VP) and (VP) and S13 is a codimension 3 singularity, (VP) S13(*) if and only if there exists a curve (V(t), P(t)) in S2 x M2, 0 t , such that P(t) S13,0(V(t)) for t > 0, and (V(0), P(0)) = (VP). 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" 1,1S1,1(*) -> S2, the restriction of the projection S2 x M2 -> S2.

Proposition 7.9 (Gaffney and Ruas) Let XM2 -> R3 be an immersion, with M orientable. If X C then

(i)
S1,1(*) 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 S13(*) and the set {(VP)|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 (VP) is equivalent to (xy3 - x2y). (Note that this germ is not stable.)

Proof See [GafR].

Theorem 7.10 If XM2 -> 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 Xn such that Xn|U converges to X|U in the Whitney topology and the family n* associated to Xn is versal on U. Since N is stable on U, we can assume that, for n sufficiently large the Gauss map Nn of Xn 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 Xn has negative curvature is a disc, so S1,1(n*) is two discs. Let Dn be one of these discs. Let n be the curve on Dn which consists of points of S1,1,1(n*). Let n be the curve in Dn lying over the parabolic curve Sn. The curve n divides Dn 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 Nn.

The projection of n to U is a curve n which is smooth except perhaps at the cusp of Nn, and which consists of inflection points of asymptotic curves of Xn. Since n crosses n transversely at a single point lying over the cusp of Nn, 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 S13(n*), they must be in S13(*), 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 1S1(*) -> 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 (VP) of 1 is stable for all (VP) 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 Immk(2, 3) be the space of k-jets of immersion of R2 in R3. Since Immk(2, 3) is a Euclidean space, with coordinates the partial derivatives of order  k, we can consider algebraic subsets of Immk(2, 3), i.e. subsets defined by polynomial equations. Recall that the parabolic image curve of the immersion XR2 -> 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 Immk(2, 3), and a proper algebraic subset W of V, such that jkX(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(xy) = (xyf(xy)), f (2)2. The set V will be all k-jets of immersions X with f(xy) =1/2x2 + a04y4 + a12xy2 + a21x2y + g(xy) where g (2)3 and gyyyy(0) = gyyy(0) = gxyy(0) = gxxy(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

Now the curvature of the parabolic image curve of X is determined by j2(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 jkX(0) V, then the curvature of the parabolic image curve of X at 0 is nonzero if and only if (a12)2 - 3a04 0

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 XM2 -> R3, is easily obtained by considering the space of 4-jets of immersion Imm4(M2R3) as a fiber bundle over M, with fiber the space of 4-jets at zero of immersions of R2 in R3.

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