In this thesis we introduce volume comparison theorem that was ex-
tended by two dierent ways: one is represented by P. Petersen and G. Wei, which
is by using Jacobian and obtains the volume comparison[P-W]; and the other is rep-
resented by G. Wei and Will Wylie, they obtain the volume comparison by using
Bakry-Emery Ricci tensor[W-W].

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