Before we go much further with the hard stuff, let's see if there is any way to know whether we are constrained to such a curved or ``warped'' [hyper]surface.
For the ``bug on the balloon'' there certainly is: simply check whether
Euclidean geometry (trigonometry, ) works properly on figures in the ``plane''
of the Earth's surface. As an extreme example, note that two ``straight
lines'' which cross at one point on the Earth will cross again
on the other side! Also note that one can make a ``triangle'' out of
great circles in which all three angles are 90o!
[Just make the length of each side equal to 
the circumference of the Earth.] And so on.