theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1437 · p. 557

Rests on

depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
- depends_on equation 14.28 eq:lie-expso2 ¶
- depends_on equation 14.25 eq:lie-so2-matrix ¶
- depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
  - depends_on definition 14.26 The rotation group ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
depends_on proposition 14.36 Closed form of the exponential ¶
- depends_on equation 14.28 eq:lie-expso2 ¶ ↺
- depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶
  - depends_on definition 14.32 The special unitary group in two dimensions ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-13 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-14 ¶
depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶ ↺
depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶
- depends_on definition 14.32 The special unitary group in two dimensions ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-11 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-15 ¶

Supports

depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
- depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
  - depends_on remark 29.12 The coordinate singularity, and what it is not ¶
- depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Rodrigues formula; the exponential map is onto	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1451
`depends_on`	→	Closed form of the exponential	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1451
`depends_on`	→	Generators of $\mathfrak{su}(2)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1451
`depends_on`	→	$\SU(2)$ is the three-sphere	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1451
`depends_on`	←	$\SU(2)$ is the universal cover	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1540
`depends_on`	←	$\Phi$ is a two-sheeted covering	declared	appendices/A-long-proofs.tex:17757
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-15	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1454