Is a uniquely geodesic space contractible? II

algebraic-topology general-topology functional-analysis metric-spaces geodesic

Solution 1:

The converse is not true. Look at the closed upper hemisphere of the 2 sphere. It is homeomorphic to a disk and this contractible. However, there are many geodesics between the two poles (for example the two different directions around the equator).

Related

Periodicity of a Function Given the Functional Equation $f(x+a)=\frac12+\sqrt{f(x)-\big(f(x)\big)^2}$ [closed]
Idempotent of closure and other properties
A Series with an increment in its difference .
If $a \in \Bbb Z$ is the sum of two squares then $a$ can't be written in which of the following forms? [duplicate]
Proving: $f$ is injective $\Leftrightarrow f(X \cap Y) = f(X) \cap f(Y)$ [duplicate]
integral involving invertible functions
What values of $0^0$ would be consistent with the Laws of Exponents?
Prove that $0 \le \frac{1+\cos\theta}{2+\sin\theta} \le \frac{4}{3}$ for all real $\theta$.
Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
Is there the exclusive list of Pythagorean triples? If there is, what is the general pattern for it?
Power summation of $n^3$ or higher [duplicate]
Cauchy integral of $\frac{1}{z}$ over closed curve