w e I(M) if there exists a manifold V (not necessarily compact, with or without boundary) and an imbedding sp: V. M (not necessarily proper) such that: (1).
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Amer. Math. Soc. translation from Dokl. Akad. Nauk. SSSR, 151, 1286-1287 (1963). 3 This theorem would be a consequence of a theorem stated without proof by Proizvolov in ref. 2. However the latter theorem is now known to be incorrect, as is also Theorem 1 of ref. 2. See an abstract by Kenneth Whyburn in Amer. Math. Soc. Notices, 11, 664 (1964). 4 Whyburn, G. T., "Open and closed mappings," Duke Math. J., 17, 69-74 (1950).
THE SINGULARITIES OF INTEGRABLE STRUCTURALLY STABLE PFAFFIAN FORMS* BY IVAN KuPKA INSTITUTO DE
MATEMXTICA
PURA E APLICADA, RIO DE JANEIRO, BRAZIL
Communicated by S. Lefschetz, September 30, 1964
Introduction.-Given a C compact manifold M, it is a now classical result (see ref. 1) that any structurally stable C X field X on M has only a finite number of singular points and that these are of generic type: this means that at such a point the differential of X has only eigenvalues with non-null real part. In the neighborhood of such a point the structure of trajectories is well determined (see refs. 2 and 3). In this small abstract we announce our generalization of this result to the dual case, that is, the case of integrable structurally stable C Pfaffian forms (see definition below). An old theorem of H. Hopf (see ref. 4) says that any field on a C manifold can be approximated by a CO field having only a finite number of singular points all of generic type. The generalization of this result to integrable Pfaffian forms seems difficult, as easily constructed examples show. A later note of ours will handle this case. Notations and Results.-M will be a CO compact n-dimensional manifold with a Riemann C --metric. T,(M) will be the tangent space of M at x E M. T(M) = U xeMTx(M) will be the fiber space of tangent vectors over M. TX*(M) will be the space of cotangent vectors at x e M and T*(M) = UXeMTx*(M) the fiber space of cotangent vectors over M. Definition 1: A CO P ffian form w on M is a C" section of T*(M). w(x0) e Tx*(M) is the value of w at xo e M. Definition 2: A C Pfaffian form w is called integrable if w A dw = 0 identically. Definition 3: A point xo e M is called singular point of a Pfaffian form w if w(x0) = 0. We call I (M) the subspace of all C- integrable Pfaffian forms with the Cl-topology. N.B.-I(M) is not a linear space. PROPOSITION 1. If M' is another C- compact manifold and if f:M' -- M is a C"- mapping, then f* I (M) = I (M'). Proof: Well known. Definition 4: A subspace D of M will be called an integral manifold of a CO- form w e I(M) if there exists a manifold V (not necessarily compact, with or without boundary) and an imbedding sp: V M (not necessarily proper) such that: (1) (p(V) = D; (2) (p*w = 0 identically (so and V being of class C1 at least). -
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Definition 5: A CO- function g on an open set Q C M is called a unit on Q if g(x) $ Oforx eQ. PROPOSITION 2. If w e I(M) and g is a CO function on an open set Q, then gw E I(Q) and any integral manifold of w is integral manifold of gw. If g is a unit on Q, then w and gw have the same integral manifolds in Q. Proof: Trivial. Definition 6: A form w e I(M) is called structurally stable if for every e > 0 there exists a neighborhood VUf of w in I(M) with the Cl-topology such that: for any w' e "Ufthere exists a homeomorphism h: M -- M with the following properties: (1) dist(x,h(x)) < e for all x E M; (2) if D is an integral manifold of w, then h(D) is an integral manifold of w'. We can now announce our main result. MAIN THEOREM. If we I(M) is structurally stable, the set of all singular points of w is the union of the following two sets: (1) A finite set 2n(w) such that if xo E Zn(w), then xo is isolated in the set of all singular points of w and there exists a neighborhood No of x0 in M and two C- functions f,g :No -- R such that: (i) g is a unit in No and f admits in No a unique singular point xo which is generic (in Morse's sense p. 172 of ref. 5) in No; (ii) w = gdf. The index of xo for f (in Morse's sense p. 143 of ref. 5) is called index of w. It determines the structure of the foliation defined by w in the neighborhood of xo. (2) A finite union 22(w) of CO- compact manifolds of codimension 2, Wi, . .. , W, such that if D is a C 2-dimensional cell transversal to Wi at a point xo, the restriction of w to D (i.e., the intersection of the foliation defined by w with D) has at xo a generic singular point. More precisely, there exists for each xo E W, a neighborhood No of xo in M, a C2 unit go:No -- R, and a C2 mapping (Po:No -O R2 everywhere of rank 2 such that if (Q,n) denotes the canonical system of coordinates in R2, ao is the form dt- X (dt (X $ 0,1 scalar), and a, is the form Wd + -jd- + Mi (qdd4ds) (Au $ 0 scalar), then either wJNo = gosoo*ao or wiNo = go'po*aj; X or ju depend only on the manifold Wj, not on the point xo e W. Except when X is rational,
