Orientability of the Sphere: A Scientific Exploration

[mritunjay]

Is the atlas for the sphere consisting of the stereographic projections from north and south pole is an orientable atlas.

 

[tiny-tim]

hi mritunjay!

what is the definition of orientable?

(alternatively, what is the definition of non-orientable?)

 

[math6]

a sphere is orientable if and only if it admits a volume form. If you're in a Riemannian manifold then the volume form is well known. Or you can use the definition of orientability of a manifold, a manifold is orientable if \ det (\ frac {\ partial x ^ {i}} {{partial y ^ {i}})> 0 for two coordinate system x ^ {i} and y ^ {i} on the variety.