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Unformatted text preview: Differential Geometry 2 Homework 04 1. Let V, ( •|• ) be an inner product space and let f be the real-valued function on V defined by v ∈ V ⇒ f ( v ) = 2 / (1 + ( v | v )) . Let g be the standard Riemannian metric on V and let g = f 2 g be the conformal metric. Explicitly calculate the curvatures R,S,s , and K . 2. Let ( M,g ) be a Riemannian manifold of dimension strictly greater than two. Assume that K (Π) has the same value for each plane Π ⊂ T p M ; show that it is then independent of p ∈ M . [The differential Bianchi identity should be considered.] Solutions (1) This can be a little lengthy; some organization helps. Let ∇ be the L-C connexion associated to g and ∇ that associated to g . Let E denote the Euler vector field on V : thus, if a ∈ V then E a = τ a ( a ) in standard notation. Direct calculations verify that f has gradient ∇ f =- f 2 E and that if ζ ∈ Vec V is any vector field on V then ∇ ζ E = ζ....
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
- Spring '09