I was given the following question in my differential geometry class. The instructor does not use a textbook, and gives only theorems and proofs with no examples, so I don't know how to do computations. Here is the problem:
Let $g = (1-x^2-y^2-z^2)^{2c}(dx \otimes dx + dy \otimes dy + dz \otimes dz)$, where $c$ is some real number, be a Riemannian metric on the open unit ball $B = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 <1 \}$. What values of $c$ makes this metric geodesically complete?
Can someone explain how to do this question? Thank you.
Hint: Consider the Riemannian metric on $(-1, 1)$ with arc length element $$ ds = (1 - x^{2})^{c}\, dx. $$ For what $c$ is the distance from $0$ to an end of the interval infinite?