Mikhael Gromov

Manifolds of nonpositive curvature

Preface.-Introduction.-Lectures on Manifolds of Nonpositive Curvature.-Simply Connected Manifolds of Nonpositive Curvature.-Groups of Isometries.-Finiteness theorems.-Strong Rigidity of Locally Symmetric Spaces.-Appendix 1. Manifolds of Higher Rank.-Appendix 2: Finiteness Results for Nonanalytic Manifolds.-Appendix 3: Tits Metric and the Action of Isometries at Infinity.-Appendix 4: Tits Metric and Asymptotic Rigidity.-Appendix 5: Symmetric Spaces of Noncompact types.-References.-Subject Index.

Analytic manifolds of nonpositive curvature

Is it possible to generalize the theorems proved for manifolds of negative curvature to manifolds of nonpositive curvature? Let us first look to the Margulis-Heintze theorem. There is an obvious counterexample: Let V’ be a compact manifold with curvature -1 ≤ K ≤ 0, then define V: = V’ × S € 1 , where S € 1 is the circle of length 2π€. Then also V satisfies -1 ≤ K ≤ 0, but the injectivity radius is everywhere smaller than €π. Thus to prove a generalization one has to exclude these products. For example we can assume that the universal cover X of V has no Euclidean de Rham factor. But also in this case, there is a counterexample: Take a Riemann surface of genus g ≥ 1 with one cusp, cut it off, then there is a Riemannian metric on this surface with curvature -1 ≤ K ≤ 0 which is isometric to S € 1 × [0,1] at the boundary. Cross this surface with S € 1 and glue it together with a second copy of itself by interchanging the factors. Thus globally the resulting manifold is not a product and the universal covering has no Euclidean de Rham factor, but the injectivity radius is small everywhere.

Bieberbach groups and a proof of the Margulis Lemma

We will describe groups operating on the Euclidean space ℝn. The isometry group of (ℝn is the semidirect product Iso(ℝn) = 0(n) ⋉ ℝn, where 0(n) is the orthogonal group. An element (A,a) ∈ Iso(ℝn) acts by (A,a)x = Ax + a, thus (A,a)(B,b) = (AB,Ab + a) and ρ: Iso(ℝn) → 0(n), ρ(A,a) = A is a homomorphism. The orthogonal map A is called the rotational part, a the translational part of (A,a). An isometry is called a translation, if it has the form (E,a) where E is the identity. We identify the translational subgroup with ℝn. By 6.7, every γ ∈ Iso(ℝn) is semisimple and MIN(γ) is an affine subspace of ℝn. An isometry γ is a translation if and only if MIN(γ) = ℝn. A discrete group Γ of Iso(ℝn) is called crystallographic, if ℝn/Γ is compact. The main result about crystallographic groups is the following well known theorem.

Local geometry and convexity

We start with an intuitive description of what the sign of the sectional curvature in a Riemannian manifold describes locally. Let us consider two geodesic rays starting from the same point p in a Riemannian manifold V and let α be the angle between these rays. If the sectional curvature K of V is everywhere nonnegative (K ⩾ 0), then the geodesics tend to come together compared with two corresponding rays (also with angle α) in the Euclidean plane, while K ⩽ 0 forces the geodesies to diverge faster than in the Euclidean situation: Open image in new window To be more precise: let V be an n-dimensional Riemannian manifold, p ∈ V and r > 0 small enough such that expp: Br(0) → Bp(p) is a diffeomorphism, where Br(0) is the open ball of radius r in the tangent space TpV and Br(p) the corresponding distance ball in V.

Special groups of isometries

We now analyze the action of some special groups of isometries acting on a Hadamard manifold X or more generally on a convex subset M ⊂ X. A group Γ operates discretely if for any compact set K ⊂ X there are only finitely many γ ∈ Γ with γK ∩ K ≠ φ. Γ operates freely (fixed point free), if γ does not contain nontrivial elliptic elements.

Rigidity and extensions of isometric maps

We have seen in the first paragraphs of this lecture that the condition K ⩽ 0 often gives us inequalities and further equality implies a strong rigidity statement.

Mostow’s rigidity theorem and its generalization. An outline of the proof

Let V* and V be compact locally symmetric spaces of nonpositive curvature with isomorphic fundamental group. Hence V* = X*/Γ*, V = X/Γ, where X* and X are symmetric spaces and Γ* is isomorphic to Γ. Let us assume that in the de Rham decomposition of X* and X there are no Euclidean factors and no factors isometric to the hyperbolic plane, then by the famous rigidity theorem of [Mostow, 1973], V* and V are isometric up to normalizing constants. Thus, if the metric of X is multiplied on each de Rham factor by a suitable constant, then X*/Γ* and X/Γ are isometric.

Proof of the rigidity theorem

Let V* = X*/Γ* and V = X/Γ be as in Theorem 14.3. By our assumption there exists an isomorphism θ. Γ* → Γ. Therefore V* and V are homotopy equivalent. Let f: V* → V and ḡ: V → V* be maps such that ḡ∘f and ḡ∘f are homotopic to the identities on V* and V. Let f: X* → X and g: X → X* be the lifts to the covering spaces. Then there are constants l,b > 0 such that f and g are (l,b)-pseudoisometries. Clearly f(γ*x*) = θ(γ*)f(x*) for x* ∈ X* and γ* ∈ Γ*, and g(γx) = θ -1(γ)g(x) for x € X and γ ∈ Γ. Furthermore there is a constant A > 0 such that d(x*,gfx*) ⩽ A and d(x,fgx) ⩽ A for x* ∈ X* and x ∈ X.

Discrete groups of isometries and the Margulis lemma

Let V be a complete Riemannian manifold of nonpositive curvature. Let X be the Riemannian universal covering and Γ ≃ κ1(V) the group of deck transformations. Then Γ is a discrete group of isometries acting freely on X and we can identify V with the quotient X/Γ. On the other hand, if Γ is a discrete group which acts freely on a Hadamard manifold X, then X/Γ is a complete manifold of nonpositive curvature. Let κ: X → V be the canonical projection, let p ∈ V and x ∈ X be a point with κ(x) = p. If c: [a,b] → V is a geodesic loop at p = c(a) = c(b), then let c: [a,b] → X be the lift of c with c(a) = x. Clearly c(b) = γx for an element γ ∈Γ. Thus the geodesic loops at p correspond bijectively to the geodesic segments from x to γx, γ ∈Γ. Because the norm ∣J∣ of Jacobi-fields is convex (§1), there are no conjugate points in V and hence the injectivity radius is half the length of the shortest geodesic loop at p. (Compare [Cheeger-Ebin, 1975] p. 95). Thus, if we define 1 , then d Γ (x) = 2 Inj Rad (κ(x)), where Inj Rad is the injectivity radius on V = X/Γ. A similar discussion shows that the convexity-radius at κ(x) is equal to 1/4 d Γ (x).

The theorem of Hadamard-Cartan and complete simply connected manifolds of nonpositive curvature

Let V be a complete Riemannian manifold of nonpositive curvature and p ∈ V. Then expp is defined on the entire tangent space TpV. The convexity of the norm of every Jacobi field (see 1.3) implies that no (non-zero) Jacobi field along a geodesic c: [0,1] → V vanishes at c(0) and c(1). Hence there are no conjugate points, and the differential of expp is everywhere nonsingular. Thus we can define a new metric on TpV, such that expp is a local isometry. This metric is complete by the Hopf-Rinow theorem, because the lines through the origin in TpV are geodesics in this metric. Now it is easy to prove that a local isometry φ: V1 → V2, where V1 is complete, is a covering map (comp. [Cheeger-Ebin, 1975], p. 35).