Proving completeness via the exponential map

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Problem 3.3.40 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 U⊂R2U \subset \mathbb {R}^{2} be an open set.

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(a)

Define a Riemannian metric on UU.

(b)

In terms of your definition, define the distance between two points p,q∈Up,q \in U.

(c)

Let Δ={(x,y):x2+y2<1}\Delta = \{ (x,y) : x^2+y^2<1\} be the open unit disc in R2\mathbb {R}^{2}, and consider the metric on Δ\Delta given by

ds2=dx2+dy2(1−x2−y2)2. ds^2 = \frac{dx^2+dy^2}{(1-x^2-y^2)^2}.

Show that Δ\Delta is complete with respect to this metric.

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.