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Tag Archives: manifold
SO(3) diffeomorphic to RP^3
} Proof: We consider as the group of all rotations about the origin of under the operation of composition. Every nontrivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. We … Continue reading
SU(2) diffeomorphic to S^3 (3sphere)
(diffeomorphic) Proof: We have that Since , we may view as Consider the map It is clear that is welldefined since if , then . If , it is clear that . So is injective. It is also clear that … Continue reading
De Rham Cohomology
De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined. Also, do check out our presentation on the relation between De Rham Cohomology and physics: De Rham … Continue reading
CP1 and S^2 are smooth manifolds and diffeomorphic (proof)
Proposition: is a smooth manifold. Proof: Define and . Also define by and . Let be the homeomorphism from to defined by and define by . Note that is an open cover of , and are welldefined homeomorphisms (from onto … Continue reading
Tangent space (Derivation definition)
Let be a smooth manifold, and let . A linear map is called a derivation at if it satisfies The tangent space to at , denoted by , is defined as the set of all derivations of at .
Smooth/Differentiable Manifold
Smooth Manifold A smooth manifold is a pair , where is a topological manifold and is a smooth structure on . Topological Manifold A topological manifold is a topological space such that: 1) is Hausdorff: For every distinct pair of … Continue reading
RP^n Projective nspace
Define an equivalence relation on by writing if and only if . The quotient space is called projective space. (This is one of the ways that we defined the projective plane .) The canonical projection is just . Define , … Continue reading
Equivalence of C^infinity atlases
Equivalence of atlases is an equivalence relation. Each atlas on is equivalent to a unique maximal atlas on . Proof: Reflexive: If is a atlas, then is also a atlas. Symmetry: Let and be two atlases such that is also … Continue reading
Tangent Space is Vector Space
Prove that the operation of linear combination, as in Definition 2.2.7, makes into an dimensional vector space over . The zero vector is the infinitesimal curve represented by the constant . If , then where , defined for all sufficiently … Continue reading
Multivariable Derivative and Partial Derivatives
If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique. Proof: Let , then becomes since . By choosing (all zeroes except in th position), … Continue reading