RIEMANNIAN MANIFOLDS WITHOUT CONJUGATE POINTS: A LECTURE ON A THEOREM OF HOPF AND

GREEN

RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA

                             HOWARD@MATH.SC.EDU

1. Introduction

In 1948 E. Hopf [3] proved that any Riemannian metric on the two dimensional torus that is without conjugate points is a flat metric. The proof proceeds by showing that any metric on a compact surface without conjugate has non-positive Gaussian curvature and then using the Gauss-Bonnet theorem to conclude that when the surface is a torus that the Gaussian curvature is identically zero. In 1958 L. Green generalized Hopf’s argument to show that any metric on a compact Riemannian manifold of any dimension that is without conjugate points has non-positive scalar curvature. The note here is based on Green’s paper gives an elementary exposition of the Hopf-Green result, however the proof is just a reworking of Green’s proof with no changes of substance.

2. Systems of Ordinary Differential Equations without Conjugate Points

In this section t ↦ R(t) will be a smooth map form the real numbers R into the vector space of m × m symmetric matrices. For any a ∈ R let S(t; a) be defined by the initial value problem

S′′(t; a) + R(t)S(t; a) = 0, S(a; a) = 0, S′(a, a) = I (2.1)

where I is the m×m identity matrix. We say that R(t) is free of conjugate points iff of all a ∈ R and t ≠ a we have det S(t; a) ≠ 0. If R(t) is free of conjugate points, then for t ≠ a define

A(t; a) = −S′(t, a)S(t, a)^{−1}.

If we view (2.1) as the Jacobi equations along a geodesic γ(t) in a Riemannian manifold then the condition that R(t) is free of conjugate points is exactly that the geodesic is free of conjugate points in the usual sense. If t < a then A(t; a) is the second fundamental form (viewed as a (1, 1) tenser) of the geodesic sphere centered at γ(t) and passing through γ(t) with respect to the normal d/dt. The horosphere determined by this geodesic is objected obtained by taking the geometric limit of these geodesic spheres as a → ∞. The following result more or less says that these second fundamental form of these horospheres exits and that it satisfies the correct matrix Riccati equation.

Date: November 1994.

Theorem 2.1 (E. Hopf [3] (m = 1) and L. Green [2] (m ≥ 2)). If R(t) is free of conjugate points then

U(t) := lim_{a→∞} A(t; a)

exists for all t, the function t ↦ U(t) is smooth and satisfies the Riccati equation

U′(t) = U(t)^{2} + R(t).

If A and B are symmetric m × m matrices then A ≤ B means that B − A is positive semi-definite. Likewise A < B will mean that B − A is positive definite.

Lemma 2.2. Under the hypothesis of the theorem, if t < a < b, then A(t; a) > A(t; b).

Proof. We first note that if A(t, a) and A(t; b) are the second fundamental forms of the geodesic spheres centered at γ(a) and γ(b) and through γ(t) then the triangle inequality implies the geodesic sphere centered at γ(b).

This can be translated into the desired inequality.

[figure: γ; γ(t)]

(If two hypersurfaces are tangent at a point and one lies on one side of the other, then there is an inequality between the second fundamental forms.) To give an analytic proof we first note that a direct calculation shows that A(t; a) satisfies a Riccati equation A′(t; a) = A(t; a)^{2}+R(t). Also from the initial value problem defining S(t, a) near t = a

S(t; a) = (t − a)I + O(t − a)^{3},
S(t; a)^{−1} = \frac{1}{(t − a)}I + O(t − a)^{3},
S′(t; a) = I + O(t − a)^{2}.

Thus

A(t; a) = −S′(t; a)S(t; a)^{−1} = \frac{−1}{(t − a)}I + O(t − a). (2.2)

For t just a little smaller than a we thus see that A(t; a) is of approximately of the form CI for C large and positive. As a < b this implies A(t, a) > A(t, b) for t just a little smaller than a. But then the comparison theory for the Riccati equation [1, Sec. 3] implies A(t, a) > A(t, b) for all t < a.

Lemma 2.3. With the hypothesis of the theorem, if a, b > 0 then A(t; a) > A(t; b) for −b < t < a.

Proof. Geometrically if γ(t) is a line, that is if it minimizes the distance between any two of its points, and A(t, a) and A(t; b) are the second fundamental forms of the geodesic spheres centered at γ(a) and γ(b) and through γ(t) then the triangle inequality implies the geodesic sphere centered at γ(a) is outside the geodesic sphere centered at γ(b). As in the last lemma this implies an inequality between the second fundamental forms.

Analytically we again use equation (2.2). If a is

[figure: γ; γ(t)]

replaced by −b in (2.2) then for t just a little larger than −b we see that A(t, −a) is approximately −CI for C a large positive constant. Thus for t just a little larger than −a we have A(t; −a) < A(t; b) and thus the comparison theory implies A(t; −a) < A(t; b) for −a < t < b.

Proof of Theorem 2.1. Fix c > 0 and let a > c. Lemma 2.3 implies that on the interval [−c, c] we have A(t, a) > A(t, −2c). Thus for some constant (only depending on R(t) and c) there holds −CI ≤ A(t, a) for all t ∈ [−c, c] and a > c. By Lemma 2.2 there for fixed t ∈ [−c, c] A(t, a) is a decreasing function of a. As there is a lower bound, we see that U(t) = lim_{a→∞} A(t; a) exists for all t ∈ [−c, c]. For t ∈ [−c, c], a > 2c we see that A′(t; a) = A(t; a)^{2} + R(t) stays bounded, so A(t; a) is uniformly Lipschitz and thus by Ascoli’s theorem the convergence in the limit is uniform. This in turn implies that as a → ∞ that A′(t; a) = A(t; a)^{2} + R(t) converges uniformly to something, and it is easy to see that this something must be U′(t). Thus U′(t) = U(t)^{2} + R(t). As U(t) satisfies a an ordinary differential equation it is a smooth function.

3. Riemannian Manifolds without Conjugate Points

Let (M, g) be a compact n dimensional Riemannian manifold without conjugate without conjugate points. Let S(M) be the unit sphere bundle of M and let ζ^{t} be the geodesic flow on S(M). Then, as usual, the geodesic flow preserves the natural volume measure on S(M). For each u ∈ S(M) let γ_{u}(t) be the geodesic fitting u (that is γ′_{u}(0) = u). Then as γ is without conjugate points we can construct a field linear maps U(t) along γ so that for each t U(t) is a selfadjoint linear map γ′_{u}(t)^{⊥} that satisfies U′(t) = U(t)^{2} + R(t) where R(t) is defined by R(t)X := R(X, γ′_{u}(t))γ′_{u}(t) and R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z is the curvature tensor of (M, g). Define a function U on S(M) by U_{u} := U(0). Thus if M is simply connected U_{u} is the second fundamental form of the horosphere determined by γ_{u} through the base point of u. For u ∈ S(M) the function t ↦ U_{ζ^{t}u} is smooth, but I do not know if the dependence of U_{u} on u is continuous, but I assume that it is not. However the map u ↦ U_{u} is measurable (in [2] Green refers us to Hopf’s paper [3] which I have yet to look at. But as far as I am concerned all functions that come up in geometry problems are measurable.)

We now take traces of the differential equation U′ = U^{2} + R and use the invariance of the Liouville measure under the geodesic flow.

0 = ∫_{S(M)} tr(U_{ζ^{1}u} − U_{u}) du  invariance of du
= ∫_{S(M)} ∫_{0}^{1} tr(U′_{ζ_{t}u}) dt du
= ∫_{S(M)} ∫_{0}^{1} tr(U_{ζ^{t}u}^{2} + R_{ζ^{t}u}) dt du
= ∫_{S(M)} tr(U_{u}^{2}) + tr(R_{u}) du  invariance of du
= ∫_{S(M)} tr(U_{u}^{2}) + Ric(u, u) du  definition of Ric
= ∫_{S(M)} tr(U_{u}^{2}) du + \frac{Vol S^{n−1}}{n} ∫_{M} Scal dx

where at the last step we have used that ∫_{S^{n−1}} Ric(u, u) du = \frac{Vol S^{n−1}}{n} Scal where Scal is the scalar curvature of (M, g). This gives the formula

∫_{M} Scal dx = −\frac{n}{Vol S^{n−1}} ∫_{S(M)} tr(U_{u}^{2}) du.

This implies at once that if (M, g) is compact without conjugate points and then the integral of Scal is non-positive and if ∫_{M} Scal dx = 0 then U_{u} = 0 almost everywhere. This implies R_{u} = U′_{u} −U_{u}^{2} = 0 for almost all u ∈ S(M). Therefore M must be flat. When n = 2 this is due to E. Hopf [3] and for n ≥ 3 it is due to L. Green [2]. Note that if n = 2 and M is a torus then by the Gauss-Bonnet theorem ∫_{M} Scal dx = 0. Thus a metric without conjugate points on a torus is flat.

References

1. L. Andersson and R. Howard, Comparison and rigidity theorems in semi-Riemannian geometry, Communications in Analysis and Geometry 6 (1998), no. 4, 819–877.

2. L. Green, A theorem of E. Hopf, Michigan Math. Jour. 5 (1958), 31–34.

3. E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 47–51.