Josai Mathematical Monographs vol 3 (2001), pp. 21-22

ON DIFFEORMORPHISMS OF A HYPERBOLIC MANIFOLD

GOTO MIDORI

1. Introduction

Let M be a complete connected Riemannian manifold of dimension n. We denote by d(x, y) the distance between two points x and y of M, and by i(x) the injectivity radius at x. If d(x, y) < i(x), then there exists a unique geodesic connecting x and y. Hence, if f is a diffeomorphism of M that satisfies the condition

d(x, f(x)) < i(x), for all x ∈ M, (*)

then f is smoothly homotopic to the identity.

We shall investigate when a diffeomorphism of M that satisfies (*) is smoothly isotopic to the identity. We will have the following theorem:

Theorem 1. Let M be a compact, connected and oriented Riemannian manifold with constant curvature -1, and f an orientation-preserving diffeomorphism that satisfies the condition (*). Set W (x) = exp_{x}^{−1}f(x). If max |∇_{v}W | ≤ 1 holds for any x ∈ M and any unit tangent vector v at x, then f is smoothly isotopic to the identity.

As an immediate consequence , we have

Corollary 1. Let M be as in the theorem. Then the group of orientation-preserving diffeomorphisms of M is locally contractible.

The local contractibility of the group of diffeomorphisms of hyperbolic spaces has been studied by Earle and Eells for 2-dimensional case, and by Hatcher for 3-dimensional case, cf. Thurston[3], in different ways.

2. Outline of the proof

Let M be a complete, connected smooth Riemannian manifold of dimension n(≥ 2). The readers refer to the book [1] for Riemannian geometric materials. Letf be a diffeomorphism of M that satisfies the condition (*) in the introduction. Then we have a vector field W on M defined by

W (x) = exp_{x}^{−1}(f(x))

for x ∈ M. It is well-defined since exp_{x} : { v ∈ T_{x}M : |v| < i(x)} → M is injective. Let x ∈ M be a point with f(x) ≠ x, and c : [0, 1] → M the unique geodesic connecting x = c(0) to f(x) = c(1), parametrized proportionally to arclength. For a unit tangent vector v at x, let α : (−ϵ, ϵ) → M be a smooth curve with α(0) = x and α̇(0) = v. We consider a variation of c, F : [0, 1] × (−ϵ, ϵ) × T_{x}M → M defined by F (t, s; v) = exp_{α(s)} tW (α(s)), where W (α(s)) = exp_{α(s)}^{−1}(f(α(s))). We see that

Y_{v}(t) := \frac{∂F (t, s; v)}{∂s}|_{s=0} = (dF_{t})_{x}(v)

is the Jacobi field along c. Also

\frac{∂F (0, s; v)}{∂s}|_{s=0} = v

and

\frac{∂F (1, s : v)}{∂s}|_{s=0} = (df)_{x}(v).

Since [∂/∂t, ∂/∂s] = 0, it follows that ∇ċY_{v}(0) = ∇_{v}W, where ∇ denotes the covariant derivative.

The above observation and the inverse function theorem yield the following: Let M be a complete connected Riemannian manifold with constant curvature

-1, and f a diffeomorphism of M that satisfies the condition (*). Set W (x) = exp_{x}^{−1} f(x). Suppose that max |∇_{v}W | ≤ 1 holds for any x ∈ M and any unit tangent vector v at x, where |V | denote the norm of V. Define F_{t} : M → M by F_{t}(x) = exp_{x}tW (x). Then F_{t} is a local diffeomorphism for each t, where F_{0} = identity and F_{1} = f.

The above result gives us a local information about diffeomorphisms on M. Applying this, together with topological consideration, we obtain the theorem in the introduction. The detail will appear elsewhere.

References

[1] J.Cheeger and D.G.Ebin Comparison Theorems in Riemannian Geometry North Holand, American Elsevier, 1975.

[2] C.J.Earle and J.Eells,Jr A fiber bundle description of Teichmuller theory. J. Diff. Geom. 3(1969), 19-43.

[3] W.P.Thurston The Topology and Geometry of 3-manifolds. (preprint), 1978, Princeton.