On the Local Stability of Differential Forms

In this paper we determine which germs of differential îforms on an n-manifold are stable (in the sense of Martinet). We show that when s ¥= 1 or when 4=1 and n < 4 Martinet had found almost all of the possible examples. The most interesting result states that for certain generic singularities of 1-forms on 4-manifolds an infinite dimensional moduli space occurs in the classification of the 1-forms with this given singularity type up to equivalence by pull-back via a diffeomorphism. In [4], Martinet proposed the following definition for stability of germs of differential s-forms on an «-manifold M. (Note. Throughout this paper all objects will be assumed to be C° differentiable.) Definition 0.1. Let w and w' be germs of s-forms on M at p and p' respectively. Then (w, p) and (w1, p') are equivalent if there exists a germ of a diffeomorphism /: (M, p) —► (AÍ, p') such that f*w' = w as germs near p. Definition 0.2. Let w be an s-form on M at p. Then w is stable at p if for any nbhd U of p there is a nbhd V of w (in the C topology on s-forms) such that if w' is in V, then there is a point p' in U such that (w, p) and (w', p') are equivalent germs. Clearly this definition depends only on the germ of w at p. Using this definition, Martinet constructs several examples of stable germs of forms. We shall show, using results of Martinet and Hsiung [3], that when s =£ 1 or when s = 1 and « < 4 the examples of Martinet are essentially the only examples of stable germs. The only new additions are in the case of (« l)-forms. We conjecture that in the remaining case when s = 1 and dim M > 4 Martinet's examples are the only examples of locally stable forms. We also show that there are no stable germs of s-forms for 2 < s < « 2 (Theorem 3.1). The most satisfactory case in the determination of stable germs of s-forms occurs when s = « 1 (Theorem 2.5). Here there is a reasonable theory which classifies these stable forms according to singularity type. This classification is Received by the editors July 3, 1975. AMS ÍMOS) subject classifications (1970). Primary 58A10; Secondary 57D45, 58C25. ( )This research was supported by NSF Contract GP 43524 and by the Research Foundation of CUNY, RF No. 11102 and RF No. 11105. one Copyright © 1976, American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 206 MARTIN GOLUBITSKY AND DAVID TISCHLER given by the order of contact at p of the line field Ker w with the hypersurface {dw = 0} where w is the germ of the (n l)-form at p. Clearly the classification of stable germs of differential forms is but a small part of the determination of all the equivalence classes for these forms. This larger classification problem makes its appearance when we try to show that a given form is not stable. Usually we have had to exhibit small perturbations of the original form which belong to different equivalence classes. In this regard, the most interesting examples occur in the case of 1-forms on 4-manifolds. Here we find that the various singularity types of germs of 1-forms (as described by Martinet) give rise to a rich and rigid geometric structure. What we show is that, except in the simplest cases, these singualrity types do not come close to describing the equivalence classes of germs of 1-forms. In fact, for at least two distinct types of singularities, an infinite dimensional moduli space appears in the classification of equivalence classes of forms with the same singularity type. See Proposition 4.7 and the proof of Theorem 4.11. Our order is as follows: we start with 0and «-forms, do (n l)-forms, then show that there are no locally stable s-forms when 2 < s < n 2, and end with 1-forms. First some notation. Let A* = A* (M) denote the vector space of exterior s-forms on T M where TpM denotes the tangent space of M at p. Let Di = Dp(M) denote the germs of differential s-forms on M at p, and let d: Lf(M) —► DS+1(M) denote exterior differentiation. Let w be an (s + l)-form and Va vector field on M. Then V J w denotes the s-form on M obtained by contracting w by V. Definition 0.3. An invariant of the equivalence class of s-forms is an assignment of a number, function, germ, etc. defined on some open set of germs of s-forms which is identical for any two equivalent germs. 1. «-forms and 0-forms. Let w be an «-form on M. Martinet [4, p. 144] and Hsiung [3, Theorem 2.2] show that w is stable at p iff either (a) w =£ 0 or (b) wp = 0 and (dh) =£ 0 where w = hv, h is the germ of a function mapping (M, p) —► (R, 0) and v is the germ of a volume form at p. Furthermore there exist coordinates x¡, . . . , xn on M at p such that in case (a) w = c/Xj A • • • A dxn and in case (b) w = XjcfXj A • • • A dxn. A 0-form w is just a function. Clearly if dw(p) = 0 then generically w is a Morse function, so the critical point p is isolated. For such forms, the critical value w(p) is an invariant of the equivalence class of w which is easily License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LOCAL STABILITY OF DIFFERENTIAL FORMS 207 perturbed. So if dw(p) = 0, w is not stable at p. Conversely, if dw(p) # 0, then w is stable at p. 2. (« l)-forms. Let w be the germ of an (« l)-form. The following argument is mostly due to Hsiung [3, p. 8, Theorem 2.3]. Lemma 2.1. Suppose that w(p) = 0, rAe« w is not stable at p. Proof. First note that if w were stable at p, then the zero of w at p would have to be transverse to the 0-section in A"~1(M) and thus be isolated. So if w' is a small perturbation of w at p, then w' would also have an isolated zero at some point p' near p. Hence if/were a diffeomorphism such that w = f*w' as germs at p, then f(p) = p'. Now suppose that dw(p) ¥= 0. Then there exists a unique vector field V on M such that w = V J dw. Clearly V has an isolated zero on M at p. Defining V' similarly for vv\ we see that if f*w' = w at p, then f*V = F*. So the eigenvalues of the linear part of V at p are invariants of the equivalence class of w at p. We claim that a small perturbation of w will change these eigenvalues, so that w is not stable at p. To see this, let w' = w + do where o is the germ of an (« 2)-form on M at p and do(p) = 0. Then V' = V + W where (CJdw = do. Since dw is a volume form and do is closed, the only restriction that we put on the perturbation W is that it be a volume preserving vector field (relative to dw). This means that the trace of the linear part of W at p is 0. Clearly there is a W so that the eigenvalues of V' are different from those of V. Next note that it is generically impossible for both w and dw to be 0 at p, which finishes the lemma. But in certain applications of this lemma we will have the situation where dw(p) is constrained to be zero. Even so the lemma is true. Let Í2 be any volume form on M. Then as above there is a unique vector field V on M at p such that w = V J Í2. Had we chosen another volume form Si' then the corresponding vector field V' would be a nonzero function multiple of V. In this case the eigenvalues of the linear part of V are not invariants but the various ratios of these eigenvalues are invariants. Clearly the perturbations IV are numerous enough to change these ratios. So w is not stable at p. Remark. We actually proved more than what was stated; namely if w(p) = 0, then w is not stable at p under perturbations by closed forms. Assuming that w is stable at p, we have two cases, dw(p) ^ 0 or dw(p) = 0. The first case is Martinet [4, p. 146]. Lemma 2.2. Ifw(p) =£ 0 and dw(p) ¥= 0, rAe« there exist coordinates JCj.xn on Mat p such that w = (1 + x{)dx2 A • ■ • A dx„. So we may assume that dw(p) = 0 and w(p) ¥= 0. Genericity implies that we may choose coordinates on M at p such that dw = JCjcfjCj A • • • A dxn. See Martinet [4, p. 144]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 208 MARTIN GOLUBITSKY AND DAVID TISCHLER Since w(p) ¥= 0, there is a well-defined line field near p given by Ker w. Let X = {dw = 0} = {xj = 0}. Again we have two cases: either Ker w* X at p or Ker w(p) Ç TpX. Lemma 2.3 (Martinet [4, p. 148] ). Suppose that w(p) + 0, dw(p) = 0, and Ker w ¡isX at p. Then there exist coordinates xt,. . . ,xnon M at p such that w = (1 ±x\l2)dx2 A • • • A dxn. Note. The sign of the x\\2 term is determined geometrically as follows: away from X dw is a volume form so there is a well-defined vector field V such that V J dw = w. The sign of x\\2 is + if the one parameter group of V moves away from X and is otherwise. So now we may assume that w(p) + 0, dw(p) = 0, and Ker wp Ç TpX. Let ̂ +2(A"_1) be the manifold of (« + 2)-jets of (« l)-forms on M at p. Let \v*p be the subset of ^+2(An_1) defined as follows: w is in W*, if (1) w(p) ¥= 0 and dw(p) = 0. (2) The zero of dw is generic. (3) Ker w(p) C TpX where X = {dw = 0}. (4) The order of contact of the line field Ker w with X is k 2. Lemma 2.4. R*, is a submanifold ofJpt+2(An~l) of codimension k-l. (We assume that k < « + 2.) Proof. Choose coordinates xx,..., x„ on M at p so that the line field Ker w is generated by 3/3x„. Let V be the vector field—defined off X— such that w = V J dw. Hence P" J w = V J(V J aw) = 0 off X So w = ^rfxj A • • • ^dxn_l off JT. By continuity w = vsax, A • • • A <&„_! on a nbhd of p where <^: (AT, p) —► R is C°°. By a simple change of coordinates we may assume that . Thus 3ft(0)/37;= ± 5¿/and det(<2,g)0 = ± 1. Since a =£ ± 1 we have a contradiction and no such diffeomorphism a exists. Thus when k = n + 1, w is not stable at 0. 3. s-forms where 2 < s < « 2. In this section we adapt Hsiung's theorem [3, p. 10, Theorem 2.6] that there are no infinitesimally stable s-forms (2 : W x U—► Jk(As) given by (w1, a) h+jk(a*w')p where a in ¿7 is viewed as a diffeomorphism of U via translation in the coordinates of ¿7 and /*(-)p : & —*Jk(As)p is the fc-jet extension map on sections. Let *w»(a) = 4>(w', a). Let Qw be the orbit through w of the natural action of Diffp(Af) = group of germs of diffeomorphisms mapping (M, p) —* (M, p). Let 0* = jk(&w;)p ■ Lemma 3.2. Ifw is stable, then Im $w> n 0* =£ 0 /or an^ w' in W, and any k. Proof. Let w' be in W. Since w is stable, there is a diffeomorphism /: (M, p) —► (M, p') such that (f*w') = w as germs at p. Let a = p'. Note in the coordinates on M at p, p = 0. Then w = f*w' = (~af)*a*w' as germs at License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LOCAL STABILITY OF DIFFERENTIAL FORMS 211 p. So a*w' = (f 1 ° a)*w and a*w' is in 0* . Hence $w>(a) = jk(a*w') is in 0*. "w Lemma 3.3. For k large enough, codim 0* in Jk(As) is greater than « wAe« 2 < s < n 2. (We can assume « > 4.) Proof. Let Diffk+1(M) = group of invertible (k + l)-jets on M at p. Clearly 0* is also given by the action of Diffk, + 1(M) on /k(A*) . So dim 0* < dim Diffk+1(M) = n