Summary: HOMEWORK 2, MAT 568, FALL 2010
Due: Thursday, Oct 28.
1. Let (M, g) be a Riemannian manifold and f : M M a diffeomor-
phism. Let fg be the Levi-Civita connection for the metric fg and g
the Levi-Civita connection for the metric g. Prove that
fg
= f g
,
i.e.
fg
X Y = g
fXfY.
From this, deduce that the (3, 1) Riemann curvature tensor transforms
naturally under pullback:
Rfg
= f
R.
2. Let g = 2g, where is a positive constant, so that g is a rescaling of
g. Show that
=

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

Collections: Mathematics