definition 13.58 Hypersurface

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2754 · p. 485

ground object -- no derivation owed

Rests on

depends_on definition 13.54 Embedded submanifold ¶
- depends_on definition 13.53 Immersion, submersion, embedding ¶
  - depends_on definition 13.52 Differential; pushforward ¶
    - depends_on definition 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 3.47 Bijective map ¶ ↺
      - depends_on definition 3.49 Composition map ¶
      - depends_on definition 3.50 Identity map ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
- depends_on example A.294 The sphere, made explicit ¶
- depends_on example 13.87 The $2$-sphere ¶
depends_on theorem 13.59 Regular value theorem ¶
- depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶ ↺
- depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
  - depends_on proposition A.565 Freeness makes every value regular ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
  - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
    - depends_on example A.572 Rotational reduction of the central-force problem ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
  - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶ ↺
- depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on proposition A.552 The component is a torus ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶
      - depends_on definition A.554 Actions on the torus ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
  - depends_on lemma A.555 The actions are well defined ¶
- depends_on proposition A.565 Freeness makes every value regular ¶ ↺
- depends_on theorem 24.47 Marsden–Weinstein reduction ¶ ↺

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`	→	Embedded submanifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2757
`depends_on`	←	The $2$-sphere in $\R^{3}$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2850
`depends_on`	←	Regular value theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2772