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 g0 be the standard Riemannian metric on V and let g = f 2 g0 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 ⊂ Tp M ; show
that it is then independent of p ∈ M . [The diﬀerential Bianchi identity
should be considered.] 1 ...
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- Spring '09
- Topology, Inner product space, Riemannian geometry, diﬀerential Bianchi identity