Orientability of the Sphere: A Scientific Exploration

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SUMMARY

The discussion centers on the orientability of the sphere, specifically addressing whether the atlas formed by stereographic projections from the north and south poles constitutes an orientable atlas. It is established that a sphere is orientable if it admits a volume form, particularly within the context of Riemannian manifolds. Additionally, the mathematical definition of orientability is clarified, stating that a manifold is orientable if the determinant of the Jacobian matrix of coordinate transformations is positive.

PREREQUISITES
  • Understanding of Riemannian manifolds
  • Familiarity with volume forms in differential geometry
  • Knowledge of coordinate transformations and Jacobian matrices
  • Basic concepts of manifold theory
NEXT STEPS
  • Study Riemannian geometry and its applications
  • Learn about volume forms and their significance in manifold theory
  • Explore the properties of Jacobian matrices in coordinate transformations
  • Investigate non-orientable manifolds and their characteristics
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Mathematicians, students of differential geometry, and anyone interested in the properties of manifolds and their applications in theoretical physics.

mritunjay
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Is the atlas for the sphere consisting of the stereographic projections from north and south pole is an orientable atlas.
 
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hi mritunjay! :smile:

what is the definition of orientable? :wink:

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

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