A. I. FET

ON THE PERIODIC PROBLEM OF THE CALCULUS OF VARIATIONS

(Presented by Academician S. L. Sobolev on 2 VII 1964)

1. Let (M) be a closed differentiable manifold, and let (J(l)) be a positive regular functional on (M). As is known, on (M) there exists at least one closed extremal of the functional (J) ((^{3,6})). In this note the case of a reversible functional is considered, i.e., one invariant under change of direction on the curve (l); such is, in particular, the length functional in a Riemannian metric. We shall call the functional (J) nondegenerate if all closed extremals of (J) are nondegenerate in the sense of Morse ((^{4})). Denote by (g^n) the curve obtained by traversing the closed extremal (g) (n) times ((n=\pm1,\pm2,\ldots)); we shall call a closed extremal (h) simple if there is no such closed extremal (g) that (h=g^n) ((n\ge2)). Closed extremals (h_1) and (h_2) are called geometrically distinct if there is no such (g) that (h_1=g^m,\ h_2=g_n), ((m,n) are integers). The existence of geometrically distinct closed extremals has so far been proved only under special topological and metric restrictions (the metric condition of Morse in the case where (M) is diffeomorphic to a sphere). In the general case the possibility was not excluded that all closed extremals on (M) are iterates of one (cf., for example, ((^{2}))). In this note we prove

Theorem 1. A positive regular, reversible, nondegenerate periodic problem of the calculus of variations on a closed manifold has at least two geometrically distinct closed extremals.

In the case of a simply connected (M) a more precise assertion is true:

Theorem 2. Let (M) be closed and simply connected, and let the first nontrivial homology group of (M) have dimension (j). Then on (M) there exist geometrically distinct simple closed extremals (g_1, g_2), whose Morse indices satisfy the relation (\lambda(g_1)+1=\lambda(g_2)\le j).

Since the case of a non-simply-connected (M) is considered in our work ((^{7})), Theorem 4, in what follows (M) is assumed to be simply connected.

1. Denote by (P,\ \bar P) the spaces of closed oriented (respectively, nonoriented) curves on (M), and by (L) the space introduced in ((^{6})), p. 288, of closed oriented curves with a marked point on (M). The set of one-point curves in each of these spaces will be denoted by (O). The set ((J(l)\le c)) in the space (C) is denoted by (J_c^C). We consider the singular homology groups
   [
   H_i(X \bmod Y,\ A),
   ]
   where the coefficient domain (A) is the group of integers (Z), the cyclic group of order (p), (Z_p), or the group of rational numbers (R).

The (i)-th type number of a closed extremal (h) in the space (C) over the coefficient field (A) is denoted by
[
m_{C,A}^{\,i}(h).
]

1. Let a closed extremal (g) have index (q). Then for every (g^n)
   [
   m_{P,R}^{\,q+1}(g^n)=0.
   ]
   For the proof, note that, according to Bott ((^{1})), p. 172, Theorems A, C), the index
   [
   \lambda(g^n)=\Sigma\Lambda(\omega),
   ]
   where (\omega) runs through all roots of degree (n) of (1) (from the simple connectedness of (M) follows the orientability of (g)), and (\Lambda(z)), (|z|=1), is an integer-valued function possessing the property

(\Lambda(\bar z)=\Lambda(z)). If (\lambda(g^2)-\lambda(g^1)=\lambda(-1)) is odd, then by Theorem 21 of A. S. Schwarz ((^8)) all (m_{\bar P,R}(g^n)=0). If, however, (\Lambda(-1)) is even, then from (\Lambda(\bar z)=\Lambda(z)) for any (n) there follows the evenness of (\lambda(g^n)-\lambda(g^1)), and our assertion follows from the same Theorem 21 ((^8)).

1. Suppose that on (M) there are no closed extremals of index (r). Then the group
   [
   H_r(J_{c_2}^{\bar P}\bmod J_{c_1}^{\bar P},R)=0.
   ]

It is enough to prove this assertion in two cases: I — when (c_2) is a critical value (J), (c_1=c_2-\varepsilon), where (\varepsilon) is sufficiently small; II — the segment ([c_1,c_2]) contains no critical values of (J). In case I the type number of the corresponding extremal is equal to zero, and the assertion is proved by means of the standard Morse deformations (((^4)) or ((^8)), Ch. V). In case II one can first apply to the space (P) the same arguments as in ((^8)), p. 41, and then, considering the involution (\vartheta(l)=l^{-1}) on (P), apply Conner’s theorem (((^8)), p. 11).

1. Let (g) be a closed extremal of least index (m) on (M). Then for (0\le i\le m), (\pi_i(M)=0), and for (0\le i\le m-1), (\pi_i(J_a^L)=0), where (a) is an arbitrary number smaller than (J(g)). Let (j) be the least value of (i) for which (\pi_i(M)\ne0). By hypothesis, (j\ge2); assume that (j\le m). By the Hurewicz theorem, (H_j(M,Z)\ne0). From the formula of universal coefficients there follows the existence of such a prime number (p) that (H_j(M,Z_p)\ne0). Construct a homomorphism
   [
   \varphi:H_{j-1}(L,Z_p)\to H_j(M,Z_p),
   ]
   analogous to that defined in ((^7)), p. 416, for the generalized fibration (\alpha) for (p=2) and the space of unoriented closed curves with a marked point. (\varphi) is an epimorphism, as in ((^7)), whence it follows that (H_{j-1}(L,Z_p)\ne0). Since (O) is homeomorphic to (M), from the exact sequence of the pair ((L,O)) we have
   [
   0\to H_{j-1}(L,Z_p)\to H_{j-1}(L\bmod OZ_p),
   ]
   so that the latter group is nontrivial. Now from Theorems 16, 19, 18, 20 ((^8)) there follows the existence on (M) of a closed extremal of index (\le j-1<m), which contradicts the definition of the number (m).

The assertion concerning (J_a^L) will be obtained by analogous arguments as soon as the applicability to (J_a^L) of the Hurewicz theorem has been proved. To prove this, let us show that (\pi_0(J_a^L)=\pi_1(J_a^L)=0) for (m\ge2), and (\pi_0(J_a^L)=0) for (m=1). If (\pi_0(J_a^L)\ne0), then one of the components of (J_a^L) would not intersect (O); the shortest closed curve in this component would have index 0, which is impossible for (m\ge1). Let now (m\ge2). In view of the one-connectedness of (O) and the covering-homotopy theorem applied to the fibered space (L\to P), (\pi_1(J_a^L)=0) will follow from the following two propositions (cf. item 4):

I. If (\lambda(h)\ge2), (b=J(h)), (b-d) is sufficiently small, and (W) is a sufficiently small neighborhood of (h) in (P), then
[
\pi_1(W\cap J_b^P\bmod W\cap J_d^P)=0.
]

II. If the segment ([c_1c_2]) contains no critical values (J), then
[
\pi_1(J_{c_2}^L\bmod J_{c_1}^L)=0.
]

I, with the aid of Morse deformations (((^4)), Ch. VIII; ((^8)), Ch. V), is reduced to the proof of the relation
[
\pi_1(E_T^k\bmod S_T^{k-1})=0\quad (k=\lambda(h^n)),
]
where (E_T^k) and, respectively, (S_T^{k-1}) are obtained from the (k)-dimensional ball (E^k) and its boundary (S^{k-1}) by identifying points congruent with respect to a certain finite group of transformations (T) (concerning the latter construction see ((^8)), pp. 16, 42). Let (\sigma) be the center of (E^k), and let (\beta:E^k\setminus\sigma\to S^{k-1}) be the radial projection; then for all (\tau\in T), (\beta\tau=\tau\beta), according to the construction of (T), and our assertion follows from the connectedness of (S_T^{k-1}).

II is easily proved with the aid of Morse decreasing deformations (cf. ((^6)), § 4, d).

1. Let (g) be a simple closed extremal of least index (m\ge1), (J(g)=c). From Morse theory (((^4)), Ch. III) it easily follows that for any (n), (\lambda(h^n)\ge\lambda(h)); therefore (g) has the least index also among all (not necessarily simple) closed extremals, and (m) coincides with the number so denoted in item 5. We perform a foliation with singu-

...with the properties of the sphere (S^{m+1}), as in ((^6)), Ch. IV; the fibers in this case are either circles, which can be continuously oriented, or points, regarded as one-point oriented curves. As is clear from the construction in ((^6)), the foliation (\gamma) has a secant surface. The base of (\gamma) is a certain polyhedron (P^m=P_1^m\cup P_2^{m-1}), where (P_1^m) is a manifold diffeomorphic to an open ball; (P_2^{m-1}\subset O); (P^m) is an integral cycle (L(S^{m+1})\bmod O(S^{m+1})). Each map (F:S^{m+1}\to M) naturally induces (\Phi_F:L(S^{m+1}), O(S^{m+1})\to L,O).

As we shall show, for a sufficiently small neighborhood (U(g)) and a sufficiently small (\varepsilon>0) there exists an (F:S^{m+1}\to M) such that (\Phi_{F*}(P^m)=z_m) is an integral cycle (L\bmod O) hanging on (g).

Find such a number (\nu) that (J_c^L) can be carried by a non-length-increasing Morse deformation into the manifold (M) of closed extremal polygons with (\nu) links. (J) defines on (M) a function (\psi) with critical point (g) of index (m). In some neighborhood (U(g)\cap M=V) there exist smooth surfaces (K^m, K^r), homeomorphic to balls, with boundaries (\dot K^m, \dot K^r), such that (K^m\subset V,\ K^r\subset V,\ K^m\cap(\psi\ge c)=\dot K^r\cap(\psi\le c)=g,\ g\in(\dot K^m\cup \dot K^r),\ m+r=\dim M), and, with suitable orientations, the algebraic index of intersection (K^m\times K^r=1) on (M). We may assume that (K^m\subset L), since the construction of the corresponding local secant surface (L) over (P) presents no difficulty. For sufficiently small (c-a), (K^m\subset J_a^L) and, by item 5, there exists a map (\Phi:\bar P_1^m\to J_c^L) such that (\Phi^{-1}(J='c)) is a point (q\in P_1^m), some neighborhood (W(q)) is mapped by (\Phi|_{\bar W}) diffeomorphically onto (K^m), and (\Phi(\bar P_1^m,\dot P_1^m)) is a single point (s\in O). Put (\Phi(P_2^{m-1})=s); in view of the existence of a secant surface of the foliation (\gamma), it is not difficult to construct such an (F:S^{m+1}\to M) that (\Phi=\Phi_F), and (\Phi_F) satisfies all the requirements.

We now turn to the proof of Theorem 2; as was noted in item 1, this is also sufficient for the proof of Theorem 1. Let first (m\ge1). As Schwarz showed (((^8)), Theorems 14, 14′), (H^i(\bar P(S^{m+1})\bmod O(S^{m+1}),R)=0) for (m+1\ge3,\ i\le m+1). Considering (P^m) as an integral cycle (\bar P(S^{m+1})\bmod O(S^{m+1})), we therefore have (P^m\sim_0 0) in (\bar P(S^{m+1})\bmod O(S^{m+1})), where (\sim_0) denotes weak homology. For (m+1=2), (2P^m\sim0) according to the direct construction in ((^5)), item 3. With the aid of the map (\bar P(S^{m+1})\to \bar P), induced by (F), we obtain (z_m\sim_0 0) in (\bar P\bmod O). For (m=0) put (z_0=g); by connectedness of (\bar P), (z_0\sim0\bmod O). We shall show that there exists on (M) a closed extremal (h) with type number (m_{\bar P,R}^{m+1}(h)\ne0). Indeed, suppose that there is no such extremal. From the result of item 4 it follows easily that, for arbitrarily small (\varepsilon>0), (z_m\sim_0 0) in (J_{c+\varepsilon}^{\bar P}). Applying Morse deformations, we find from this that, for some integer (\nu), (\nu K^m\sim0) on ((\psi\le c)\bmod(\psi