Manifold Construction Examples

Differential Geometry · Axiom Academy

EXAMPLE Manifold Construction Examples Step-by-step constructions of fundamental manifolds in differential geometry Excellent work! You've explored fundamental manifold constructions. Here's what we learned: Stereographic Projection: The 2-sphere S² can be covered by two charts using stereographic projection from the north and south poles, with smooth transition maps given by inversion. Product Manifolds: The torus T² = S¹ × S¹ demonstrates how product manifolds inherit their atlas from coordinate charts on each factor, with dimension equal to the sum of dimensions. Quotient Spaces: Real projective space RP n is constructed as a quotient of the sphere by identifying antipodal points, creating a manifold with half the "apparent" dimension. General Construction: Product manifolds M × N obtain their smooth structure from component atlases, with charts (φ × ψ) and transition maps that are smooth in each component. These constructions form the foundation for understanding more complex manifolds in differential geometry!

This is the written version of the interactive lesson above. See the full Differential Geometry course.