Small Perturbations and Infinitesimal Deformations on Surfaces of Revolution

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a general response of a problem started with an idea of M. Spivak.


Introduction
There are many different ways in which a geometric structure may be rigid, unwarpable, bendable or deformable. A precise formulation of any of these intuitions may be, for example, macroscopic or infinitesimal and may involve the notion of an (ambient) isotopy or not. The interest in this type of problem matured in the late part of the nineteenth century (see [11]) and since then, the vast accumulated literature on the subject has also covered all sorts of applications in Architecture, Engineering and Mechanics (see [4], [5], [6], [8]). In any event, the paper of Stefan Cohn-Vossen, published in 1930, (see [2] ) is certainly a classic approach to our main subject.
From the mathematical point of view, the partial differential equations and the analytic methods involved have been a matter of recent interest (see [12] and [13]). Despite the wealth of important results in the area, Shing-Tung Yau states, for example, that "the study of the rigidity of nonconvex surfaces is still in its infancy" (see [12, p. 27]).
The main result of this work is: Let λ = (r, h) : S 1 → R 2 be an embedding such that r(s) > 0, ∀s ∈ S 1 , and let K = λ(S 1 ). Then, for any compact set B ⊆ R 2 containing K in its interior, there is an "admissible" perturbation α : [o, 1] × R 2 → R 2 supported in B such that the surface of revolution obtained by revolving the perturbation of λ at time one by α admits a nontrivial infinitesimal deformation.
This theorem starts with an idea of M. Spivak (see [10,Vol. 5,) about the infinitesimal deformations of surfaces of revolution. Spivak works with a particular kind of surface diffeomorphic to the two dimensional sphere satisfying a restrictive analytical condition at its poles. In contrast, our methods work for general surfaces of revolution, diffeomorphic either to the 2-sphere or to the 2-dimensional torus. For this matter, the elementary part of Morse Theory is used, thus introducing a new way of combining analytical and geometric methods which allows substantial generalization and clarification. This is done in section 3, where further remarks are given.

Infinitesimal Rigidity
Definition. Let N be a manifold and let ϕ : N → R m be an embedding. A bending of ϕ is a smooth map α : for some isometry A t of the ambient space R m ; it is called nontrivial if at least oneᾱ(t) is not of this form. We say that the embedding ϕ is bendable if there is a nontrivial bending of ϕ; otherwise it is called unbendable.
N be a manifold and let T N denote its tangent bundle. For G : N → R m , let T G : T N → T R m ∼ = R m × R m be its derivative and let dG : T N → R m be the projection of T G on the second coordinate. Let U ⊆ R k be open an let G : U → R m be smooth. Denote by JG : U × R k → R m the Jacobian matrix of G. Endow R m with the Riemannian metric defined by its dot product •. If A is a matrix, denote its transpose by A t .
Definition. Let ϕ : N → R m be an embedding. A vector field Z : N → R m is an infinitesimal deformation for ϕ if it satisfies: Infinitesimal deformations arising from trivial bendings are called trivial. If ϕ admits only trivial infinitesimal deformations, then it is called infinitesimally rigid.
It is not difficult to verify the following proposition.
Proposition. Let N be a manifold of dimension k and let ϕ : N → R m be an embedding. A vector field Z : N → R m is an infinitesimal deformation for ϕ if and only if there is an atlas

Perturbations of Surfaces of Revolution
Let S 1 be the unit circle thought of as the real numbers modulo 2π (ie. s ∼ (cos s, sin s), ∀s ∈ R). Therefore we may think of a circle point as a unit complex number or as a lateral class s + 2π · Z. Let λ = (r, h) : S 1 → R 2 be an embedding such that r(s) > 0, ∀s ∈ S 1 . Set γ(t) = (cos t, sin t, 0) and k = (0, 0, 1). Let F : S 1 ×R → R 3 be given by F (s, t) = r(s)·γ(t) + h(s)· k. Therefore, S = F (S 1 × R) is a smooth surface and F is a local embedding. With some ambiguity we make the following definition.
Definition. F (λ) = F (or S) is called the surface of revolution obtained by revolving the curve λ around the vertical axis and (globally) parametrized by F .
Definitions. Let N be a topological space and set l = 0; or let N be a smooth manifold and set l = 3. A C l map α : Let G : N → N be a C l map. The closure of the set {p ∈ N | G(p) = p} is the support of G and is denoted by supp(G). For an isotopy α define its support as the closure of the set {suppᾱ(t) | t ∈ [0, 1]}.
The next statement is the main result of this work.
Theorem. Let λ = (r, h) : S 1 → R 2 be an embedding such that r(s) > 0, ∀s ∈ S 1 , and let K = λ(S 1 ). Then, for any compact set B ⊆ R 2 containing K in its interior, there is an admissible perturbation α for K supported in B such that F (ᾱ(1) • λ) admits a nontrivial infinitesimal deformation.
Definition. Let N be a manifold and let g : A critical point p is nondegenerate if, for some (hence any) coordinate chart β around p, the Hessian matrix of g • β −1 at β(p) is nonsingular. Functions whose critical points are nondegenerate are called Morse functions.
For an arbitrary oriented one dimensional manifold K ⊆ R 2 , let κ : K → R denote its (signed) curvature and let ζ : K → R be the restriction to K of the projection (x, z) → z. Lemma 1. Let K ⊆ R 2 be a compact one dimensional manifold. Suppose the height function ζ : K → R is Morse. Then (a) C(ζ) is finite, (b) κ(p) = 0 for all p ∈ C(ζ).
(c) For each p ∈ C(ζ), there is a coordinate chart β : Proof. For any Morse function the set of critical points is closed and discrete (see Hirsch [3, p. 143 is nonoriented, substitute β for β • φ, so we may always assume that β is oriented if necessary. In any case, z is unchanged. Now, showing that κ(p) = 0 q.e.d.
Then the set of ν ∈ S 1 such that f ν is a Morse function is open and dense.
Corollary. Let K ⊆ R 2 be a compact one dimensional manifold. Let R θ denote the plane rotation at angle θ in the positive direction with center at the origin and let K θ = R θ (K). Then, for almost every θ ∈ R, the height function Definition. Let K ⊆ R 2 be a one dimensional manifold with Morse height function ζ and let p ∈ C(ζ). Let x be the horizontal coordinate (x, z) → x. Then x must be a diffeomorphism around p; therefore, ζ • x −1 is locally defined. The point p is called even analytic if ζ • x −1 is a real even analytic function around x(p). K is even analytic (at C(ζ)) if all its (nondegenerate) critical points are even analytic.
be a compact one dimensional manifold. Then, for any compact set B ⊆ R 2 containing K in its interior, there is an admissible perturbation α for K supported in B and arbitrarily close to the identity such thatᾱ(1)(K) has Morse height function ζ 1 and is even analytic (at C(ζ 1 )).
The one parameter group of diffeomorphisms R : R × R 2 → R 2 induced by X is such that everyR(t) is supported in B and there exists ǫ > 0 where R(t, p) is the rotation at angle t in the positive direction with center at the origin (∀ | t |< ǫ and ∀ p ∈ K). Choose 0 < θ < ǫ such that the height function ζ θ :R(θ)(K) → R is Morse. Note that θ may be chosen as small as desired (ǫ is not necessarily small). Let By making δ 1 smaller, elementary Taylor theory says that there exist K < 0 such that By making δ 1 smaller, now we can find δ 2 > 0 such that Let η 1 : R → [0, 1] be a smooth function such that which is certainly even analytic at x 0 , and let Y be the vector field given by Note that Y is vertical and points upwards when is different than zero. The one parameter group of diffeomorphisms S : ). For (t, x) ∈ [0, 1]× < x 0 −δ 1 , x 0 +δ 1 >, (and condition (5)), we have (t, (x, ρ(x))) → (x, ρ(x)+tη 1 (x)(f (x)−ρ(x)) = (x, g t (x)).
The situation for convex critical points is treated in exactly the same way. Hence, we can perturbR(θ)(K) at all points in C(ζ θ ) (disjointly) in order to make all resulting critical points even analytic q.e.d.
Lemma 2 and the following Proposition imply the Theorem of this section.
Proposition. Let K ⊆ {(x, z) ∈ R 2 | x > 0} be diffeomorphic to the unit circle S 1 . Suppose K has Morse height function ζ which is also even analytic (at C(ζ)). Let λ = (r, h) : S 1 → K be a diffeomorphism, so the surface of revolution F (λ) is defined. Then, for any compact set B ⊆ R 2 containing K in its interior, there is an admissible perturbation α for K supported in B such that F (ᾱ(1) • λ) admits nontrivial infinitesimal deformations different from zero almost everywhere.
Proof. (cf Spivak [10, Vol. 5, pp. 253-260]). Let F = F (λ). Then, any vector field Z along F can be written uniquely as Z(s, t) = a(s, t) · γ(t) + b(s, t) · γ ′ (t) + c(s, t) · k for some smooth functions a, b, c. An infinitesimal deformation Z for F is trivial if it is of the form Z(s, t) = µ × F (s, t) + ω, where µ, ω ∈ R 3 and × denotes the cross product. Since F is a global regular parametrization, its local inverses form an atlas for its image. Hence, by the Proposition of section 2, Z is an infinitesimal deformation if an only if (JZ) t · JF is a 2 × 2 antisymmetric matrix. That is, ∂Z ∂s A simple calculation shows that these equations are explicitly given by Propose standar solutions of the form For a (complex-valued) solution involving a single k, equations ( †) become If this system is solved, then it is easy to see that a k (s, t) = e ikt φ k (s) + e −iktφ k (s), b k (s, t) = e ikt ψ k (s) + e −iktψ k (s), c k (s, t) = e ikt ξ k (s) + e −iktξ k (s) are the scalar components of a real infinitesimal deformation.
Differentiating the third equation in ( † †) and using the other two, we obtain the equivalent system Solutions of the third equation above immediately give solutions of the first two. Hence we can define infinitesimal deformations away from the critical set {F (s, t) | h ′ (s) = 0}. Now, ζ is given by (x, z) → z and if we let x to be the horizontal coordinate (x, z) → x, then r = x • λ and h = ζ • λ. Also, R is (periodically) partitioned into (a) open intervals where h ′ is nonzero and, (b) its endpoints, which correspond to the elements in C(ζ) ⊆ R 2 .
We shall refer to the closures of these intervals (or its images under h or λ) as "the segments". In particular, for any s ∈ R, if h ′ (s) = 0 then r is locally invertible around s. For such points let ρ = h • r −1 = ζ • x −1 , so ρ is even analytic. Incidentally, this proves that the number of elements in C(ζ), say ♯ | C(ζ) |, is even, because intervals where h ′ is positive and negative must alternate, for λ −1 (C(ζ)) contains no inflection points. Clearly 2 ≤ ♯ | C(ζ) |, for the top and bottom points in K are critical. Also, the equality h • r −1 = ζ • x −1 shows that if s ♭ = s ♮ + 2 · π correspond to the same point in C(ζ) then the associated ρ is the same. Observe that {s ∈ R | r ′ (s) = 0} may be rather nasty.
Define, when meaningful, Finally, the second equation of this last system may be changed to obtain the equivalent system Note that, when both r −1 and h −1 are defined, Again, solutions for the third equation in (5 − †) immediately produce solutions for the first two away from critical points. In the same manner, solutions for the third equation in (6 − †) immediately produce solutions for the first two around critical points.
Fix p ∈ C(ζ). Then, ρ is (even) analytic around x 0 = x(p) with zero derivative at x 0 . Then it can be expressed as ρ(x) = ζ(p) This is a standard expression of a second-order differential equation having a regular singular point at x 0 with roots 0 and 2 for its "indicial equation". Therefore, it has a unique analytic solution of the form Ψ k (x) = (x−x 0 ) 2 A(x), satisfying A(x 0 ) = 1, (see Birkhoff and Rota [1, p. 252 ff]). In particular, x 0 is an isolated zero for Ψ k (x). Note that x 0 > 0, but this is a geometric condition that assures that we have a smooth surface of revolution. From the analytic point of view, x 0 = 0 is admisible. In this case the indicial equation has roots 1 ± k. If k > 1, there is exactly one analytic solution (around 0) of the form Ψ k (x) = x 1+k A(x), (cf [10, Vol. 5, p. 257]).
Up to now k has been kept fixed. To solve the above mentioned obstruction we will have to vary this integer and perturb K (a third time, but now away from C(ζ)).
Let f 1 : [z 0 −2δ 2 , z 0 +2δ 2 ] →< x 0 −δ 1 /2, x 0 +δ 1 /2 > be a smooth function such that (5) f 0 and f 1 have the same values and the same first and second derivatives at z 0 ± 2δ 2 , The horizontal field X(x, z) = (f 1 (z)−f 0 (z), 0), ∀z ∈ [z 0 −2δ 2 , z 0 +2δ 2 ] and X(x, z) = (0, 0), ∀z ∈ [z 0 −2δ 2 , z 0 +2δ 2 ] is C 2 and may be damped by a bump function (in variable x) in order to produce a small flow that sends, in time one, the graph of f 0 onto the graph of 1] be the continuous family of horizontal coordinates produced by the flow (for the segment [z 1 , z 2 ]). Fix k ∈ N and consider the family of equations (for t ∈ [0, 1], on < z 1 , z 2 >) and let { Ψ k,t } t∈[0,1] be a continuous family of solutions such that its expression in the horizontal axis has R(z 1 ) as a zero of order two. Then, 1] is a family of solutions of the same differential equation in that interval. Therefore, by the Sturm Comparison Theorem, for any where, as before, ♯[V ] denotes the cardinality of set V . (11) By concavity of f 0 = R 0 in [z 0 − 2δ 2 , z 0 + 2δ 2 ], Ψ k,0 has at most one zero in the interval. (12) By concavity of f 1 = R 1 in [z 0 − 2δ 2 , z 0 − δ 2 > ∪ < z 0 + δ 2 , z 0 + 2δ 2 ], Ψ k,1 has at most one zero in each of these intervals.
Considerations (9 -12) hold for any k. Now, if k is sufficiently large, by the Sturm Comparison Theorem and the convexity of R 1 in < z 0 − δ 2 , z 0 + δ 2 >, Ψ k,1 can be made to have many zeroes in the interval. For such value of k, . We want to prove that Ψ k,t (z 2 ) = 0. Let {Υ 0,t , Υ ∞,t } t∈[0,1] be a family of linearly independent solutions of equation (♣) in < z 1 , z 2 > such that Υ 0,t (z 2 ) = 0 and Υ ∞,t (z 2 ) = 0 (possibly ±∞), for all t ∈ [0, 1]. Then there exist unique Since the Wronskian at z 0 is not zero, then This relation, and the C 1 uniform continuity in compact sets, of the state space and parameter space, of the solutions of differential equations implies that the coefficients d 0 , d ∞ are continuous. Suppose that Ψ k,t (z 2 ) = 0. Then d ∞ (t ) = 0 and certainly Υ 0,t (z 2 ) = 0. This and the continuity of the situation imply that there exist ν 1 , ν 2 > 0 and a neighbourhood U ⊆ [0, 1] oft such that ν 1 <| Ψ k,t (z) | for all t ∈ U and all z ∈ [z 2 − 2ν 2 , z 2 >.
By applying the above constructions to all segments in a given order, the Proposition is proved q.e.d.

Remarks.
(1) It is not difficult to verify that, for k ≥ 2, infinitesimal (nonzero) deformations obtained just by solving system ( † † † †) are nontrivial. Also, the "Fourier" nature of the solutions proposed says that nonzero solutions for different k ′ s are linearly independent.
(2) The "glueing" of function f 1 with f 0 in item (5) of the above Proposition may be much improved so as to make the subsequent arguments as smooth as pleased. But suppose s 1 < s 2 < · · · < s ℓ+1 , where s 1 = s ℓ+1 (mod 2π Z), is a sequence of numbers corresponding to the points in C(ζ). The continuation process through the various s j ′ s can be done as smoothly as the solutions to system (♣) are; except at the end when one finally reaches point s 1 again. No matter how smooth are the original data to the problem, in general, the glued solutions at s 1 are to be only of class C 2 (if the family { R t } t∈[0,1] is C 4 ).
(3) Any topological embedding S 1 ֒→ R 2 can certainly be perturbed by arbitrarily small isotopies resulting in a C ∞ embedding. Nevertheless, "tracking differentiabilities" may always prove to be of use in later situations (see Rado [9]).
(4) Let (K, ∂K) ⊆ ({(x, z) ∈ R 2 | x ≥ 0}, {0} × R) be diffeomorphic to the unit interval ([0, 1], ∂[0, 1]) in such a way that K intersects the vertical axis orthogonally. Then the surface of revolution obtained by revolving K is topologically a sphere subject to the methods of this work. Indeed, first consider the one-manifold K ∪M(K), where M : (x, z) → (−x, z). The analog of Lemma 2 can be obtained without difficulty as well as the succeeding Proposition. At the set of poles ∂K is where the fact that the height function is Morse even analytic, and not just Morse analytic, becomes handy. These are points of null abscissa hence the indicial equation has roots 1 ± k and therefore, the best is to start the continuation process from there. Also, is easy to imagine examples of noncompact surfaces of revolution for which our main result is true.