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 non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. We … Continue reading

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SU(2) diffeomorphic to S^3 (3-sphere)

(diffeomorphic) Proof: We have that Since , we may view as Consider the map It is clear that is well-defined since if , then . If , it is clear that . So is injective. It is also clear that … Continue reading

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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

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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 well-defined homeomorphisms (from onto … Continue reading

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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 .

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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

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RP^n Projective n-space

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

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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

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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

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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

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