Summary: HOMEWORK I, MAT 568, FALL 2010
Due: Tuesday, Oct 12.
If you've not already done so, read and understand all of Chapter 1 of
the Petersen text. Ask if you have any questions. Read also some aspects
of Chapter 5, namely Sections 5.2-5.3, 5.5 and also the proof of the Hopf-
Rinow theorem in §5.8. Either ignore concepts used in those sections we
haven't covered yet, or learn about them from the text, (e.g. the definition
of "geodesic segment").
1. Problem 1, p.17: Given Riemannian metrics gM and gN on manifolds
M, N, the product metric on M × N is the metric gM + gN .
(a). Show that (Rn, gEucl) = (R, dt2) × · · · × (R, dt2).
(b). Show that the flat square torus
, ( 1
) × (S1