CONFORMAL GEOMETRY SEMINAR The Poincaré Uniformization Theorem
نویسنده
چکیده
1.1. Geometry. A covariant derivative on a manifold M is an operator ∇XY on vector fields X and Y satisfying for any smooth function f : (i) ∇fXY = f∇XY ; and (ii) ∇X(fY ) = f∇XY + (∇Xf)Y . If g is a Riemannian metric on M , then there is associated with g a unique covariant derivative ∇ characterized by: (iii) ∇XY −∇YX = [X,Y ]; and (iv) ∇X ( g(Y,Z) ) = g(∇XY, Z) + g(Y,∇XZ). We define the Christoffel symbols by Γjk = dxi ( ∇∂j∂k ) , where ∂i is a coordinate basis, and dxi is the dual basis. The Christoffel symbols can be computed from:
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