# 3-manifold Groups and Nonpositive Curvature by Kapovich M.

By Kapovich M.

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Additional info for 3-manifold Groups and Nonpositive Curvature

Example text

2. M is a compact manifold. 3. The fundamental group of M is finite. We refer to [16] for the proofs of these theorems. 5 Parallel Displacements and Holonomy Groups In this section we will introduce the notions of parallel displacements and holonomy groups in Finsler geometry. This will be useful in the next section, where we will define the notions of Berwald and Landsberg spaces. Let (M, F) be a connected Finsler space and σ (t), a ≤ t ≤ b, a piecewise smooth curve in M connecting p and q, with velocity vector field T .

Let G be a Lie group and H a Lie subgroup of G. Then the Lie algebra h of H is a subalgebra of the Lie algebra g of G. Conversely, given any subalgebra h of g, there exists a unique connected Lie subgroup H of G whose Lie algebra is h. 4. Let G and H be two Lie groups. A map ϕ from G to H is called a homomorphism if ϕ is an abstract group homomorphism and it is continuous with respect to the topology of the groups. A homomorphism ϕ is called an isomorphism if ϕ is a homeomorphism. In case there exists an isomorphism from G onto H, we say that G is isomorphic to H.

We say that sup x∈BO (r) A x = o( f (r)) if the left side grows more slowly than f (r) as r → ∞. One can similarly define the above setting for L or other types of tensors. 14 (Akbar-Zadeh [2]). Let (M, F) be a connected Finsler space of constant flag curvature λ . Let O be a designed origin of M, and ∇ the Chern connection on π ∗ (T M) over T M \ {0}. 8 Spaces of Constant Flag Curvature and Einstein Metrics 27 1. Suppose λ < 0 and (M, F) is complete. If A or L sup x∈BO (r) x =o e √ −λ r , then (M, F) must be Riemannian.