ALMOST ALL SIMPLY CONNECTED CLOSED SURFACES ARE RIGID BY HERMAN GLUCK 1, INTRODUCTION, Are closed surfaces rigid? Euler thought so, and conjec- tured in 1766, "A closed spacial figure allows no changes, as long as it is not ripped apart" [6], and expanded on this in letters to Lagrange in 1770. But the conjecture has not yet yielded, and is surely one of the oldest and most beautiful unsolved problems in geometry in the large. What is the evidence in its favor? The experimental evidence is of two sorts. Most cardboard models of closed surfaces, such as the boundaries of the regular solids (but also nonconvex surfaces and those of different topological type) seem to be rigid and not flex. More inter- esting, some models do seem to flex, but in each case the apparent flexing has been traceable to slight distortions, such as bending of the faces or separation of the vertices, due to the nature of construction. The first mathematical advance was made by Cauchy [2] in 1813, who proved that two closed convex polyhedra, constructed from pairwise congruent faces assembled in the same order, were in fact congruent themselves. Hence a closed strictly convex polyhedron must be rigid because any slight flexing of it would still be convex and hence congruent to it. Similar results were obtained by Liebmann [8] in 1899 for analytic surfaces, and by Cohn-Vossen [3] in 1936 for the smooth case. I offer here a simple argument that a closed simply-connected surface in three space is almost always rigid; Euler's conjecture in this case is therefore "statis- tically" true. I think the same should be provable for any closed surface in three space, regardless of topological type, but I have been unable to do this. In order to provide a self-contained introduction to the rigidity problem, I have included Alexandrov's proof of the infinitesimal rigidity of strictly convex polyhedra in section 5. The reader wishing a guide to the history and literature of this problem should first consult Efimov's introduction and appendix in [5], and then the standard bible [i] by Alexandrov. 226 This paper is organized as follows: 2. Rigidity - competing definitions and their equivalence, 3. Infinitesimal rigidity - competing definitions and their equivalence, 4. Infinitesimal rigidity implies rigidity, S. Strictly convex closed surfaces are infinitesimally rigid, hence rigid, 6. Almost all simply connected closed surfaces are infinitesimally rigid, hence rigid. I have profited from reading Robin Langer's honors thesis [7], in which several of the ideas to be discussed below were recast in more elegant form, and I have bor- rowed from his presentation. I have also benefited from many discussions with David Singer. Some new and exciting work on the rigidity conjecture is being done by Robert Connelly, using methods of complex analysis, but this is not yet in print. 2, RIGIDITY, Let K be a simplicial complex whose underlying space is ho- meomorphic to the two-sphere. K is otherwise arbitrary and will remain fixed for the entire story. A polyhedron in three-space, combinatorially the same as K, is realized by a simplexwise linear map P : K + R 3 . For simplicity, we do not exclude the degener- ate maps at this point. Such maps are determined by their values on the vertices vl, v2,..., v V of K, and hence correspond to V-tuples (PI' P2' .... PV ) of points of R 3, The set of 3V all such polyhedra is therefore parametrized by R , and by abuse we allow our- selves to write P = (PI' P2 .... ' PV ) @ R3V Our goal is to show that almost all such polyhedra are rigid, in that the set of rigid ones contains an open and dense subset of R 3V. Two polyhedra P = (Pl .... 'Pv ) and Q = (ql' .... qv ) are CONGRUENT, written P ~ Q, if there is a rigid motion h : R 3 + R 3 such that hP = Q : K + R 3. Equiva- lently, II Pi - Pj II = H qi - qj II for 1 ! i, j iV , where II Jl is the Euclidean norm in R 3. The congruence class of P in R 3V will be denoted by [[P]]. By contrast, P and Q are I$OM£YRIC, written P ~ Q, if each face of P is congruent to the corresponding face of Q, equivalently if P and Q both pull the Riemannian metric on R 3 back to the same (possibly degenerate) metric on K. Let E be the set of pairs (i,j), 1 ~ i, j ~ V, for which an edge of K connects the vertices v. and v.. Then P is isometric to Q if and only if l ] 227 [I Pi - Pj II = II qi - qj II for (i,j) c E . Note that this capitalizes on the fact that the faces of K are triangular. The isometry class of P in R ~V will be denoted by [P]. Since congruence implies isometry, [[P]] c [p]. Though they are equal for a tetrahedron, they may easily be unequal. For example, if P is a house whose roof slopes on all four sides, and Q is reconstructed from the same pieces but with its roof dipping into the body so as to form a large trough, then P and Q are iso- metric but not congruent, so that [[P]] is a proper subset of [P]. What do [[P]] and [P] look like as subsets of R3V? Since the equation LI Pi - Pj ;I = II qi - qj II can be written as a quadratic polynomial in the coordi- nates of the vertices of Q, both [[P]] and [P] are algebraic varieties in R ~V. If P does not lie in any plane in R 3, then the group of all rigid motions of R 3 acts effectively on [[P]], which is therefore homeomorphic to two disjoint copies of R 3 x p~ (where p3 denotes projective 3-space), embedded as an alge- braic submanifold of R 3V. The appearance of [P] is less clear. If the rigidity conjecture is correct, then any embedded P is rigid and therefore (as we will see shortly) [P] is a finite disjoint union of congruence classes and hence an algebraic submanifold of R 3V. But a priori we only know that [P] is an algebraic variety. Indeed, if we consider instead the simplexwise linear maps of a quadrilateral into the plane R z, and let P denote a square, then [P] is homeomorphic to R z x S 1 x (S 1 V SI), where S 1 V S 1 denotes the wedge of two circles (a figure eight). Hence in this case, [P] is a four dimensional algebraic variety with singularities along a three dimensional subvariety. We now crystallize the notion of rigidity by first offering some competing def- initions and then observing that they are equivalent. Let H Ii also denote the Euclidean norm in R 3V, so that we may speak of the distance between polyhedra (each a linear map of K into R3). DEFINITION 2,1. A polyhedron P is RIGID if there exists an E > 0 such that any polyhedron Q, isometric to p and within g of P, is actually con- gruent to P. In symbols, Q ~ p and II Q - P II < c ~ Q ~ P . For the purposes of the following discussion, we also say Q is ~-RIGID. DEFINITION 2.2, P is RIGID if any path in [P] beginning at P actually lies entirely in [[P]]. 228 DEFINITION 2,3, p is RIGID if any analytic path in [P] beginning at P lies entirely in [[P]]. The first definition calls P rigid if a dissection and reassembly of P into a polyhedron approximating it can only yield a bodily displacement of P. The second calls P rigid if no continuous "flexing" of P is possible, and the third is just a technical modification of this. REMARK (2,4), If P is g-rigid and Q is congruent to P, then Q is also g-rigid, for the same c. Hence [[P]] has distance > g from the rest of [P]. REMARK (2,5), ~or any P, [P] has only finitely many topological compo- nents, since as a real algebraic variety it can be written as a finite disjoint union of connected real algebraic manifolds by [12]. REMARK (2,6), Any collection of polyhedra which are isometric to one another, pairwise noncongruent and g-rigid (for varying g) must be finite. For by (2.4), each P in the collection yields two components of [P], namely the two pieces of [[P]], while by (2.5) there can only be finitely many components. THEOREM 2,7, The three definitions of rigidity are equivalent. If P is e-rigid, then a continuous path in [P] beginning at P would, by (2.4), have to remain in [[P]]. Hence the first definition implies the second, which obviously implies the third. Suppose the first definition fails. Then any neighborhood of P in [P] con- tains points of [P] - [[P]]. Since [P] is a real algebraic variety and [[P]] a subvariety, Lemma 18.3 of [i0] can be applied to yield an analytic path ~(t) in [P], with ~(0) = P and ~(t) E [p] - [[p]] for t > 0. Thus the third definition also fails, so all three are equivalent. 3, INFINITESIMAL RIGIDITY, In this section we give two definitions of in- finitesimal rigidity, provide algebraic formulations and geometric motivations for each, and then prove they are equivalent. The notion of infinitesimal rigidity is stricter than that of rigidity, and corresponds to the conditions an engineer would require to be satisfied by a collec- tion of rods, joined but freely pivoting at their ends, before certifying rigidity.