The exponential map as a local diffeomorphism

[1/100%]
Show
LaTeX
Problem 3.3.12 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let MM be a Riemannian manifold. Let p∈Mp \in M.

?
(a)

Show that there exists δ>0\delta > 0 such that

exp⁡p:Bδ(0)⊂TpM→M \exp _p : B_\delta (0) \subset T_pM \to M

is a diffeomorphism onto its image.

(b)

Show that there exists ε>0\varepsilon > 0 such that exp⁡p(Bε(0))\exp_p(B_\varepsilon (0)) is a convex set.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.