theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain
open in the book ·
parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5169
· p. 516
Rests on
-
depends_on
definition 13.105
Closed form
¶
-
depends_on
definition 13.103
Exterior derivative
¶
-
depends_on
definition 13.98
$k$-form
¶
-
depends_on
definition 13.83
Covector
¶
- depends_on definition 13.81 Vector on a manifold ¶
-
depends_on
definition 13.84
Tensor
¶
- depends_on definition 13.83 Covector ¶ ↺
- depends_on definition 13.81 Vector on a manifold ¶ ↺
-
depends_on
definition 13.83
Covector
¶
-
depends_on
definition 13.102
Wedge product
¶
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on equation 13.239 eq:mfd-dual-basis ¶
-
depends_on
definition 13.98
$k$-form
¶
- depends_on definition 13.98 $k$-form ¶ ↺
-
depends_on
definition 13.103
Exterior derivative
¶
-
depends_on
definition 13.106
Exact form
¶
- depends_on definition 13.103 Exterior derivative ¶ ↺
- depends_on definition 13.98 $k$-form ¶ ↺
-
depends_on
definition 13.108
Star-shaped domain
¶
-
depends_on
definition 6.2
Open set
¶
-
depends_on
definition 6.1
Topological space
¶
- depends_on definition 3.31 Empty set ¶
-
depends_on
definition 3.35
Union, intersection, difference
¶
- depends_on definition 3.8 Conjunction ¶
- depends_on definition 3.9 Disjunction ¶
- depends_on definition 3.7 Negation ¶
- depends_on definition 3.30 Subset ¶
- depends_on equation 3.51 eq:set-indexed ¶
-
depends_on
definition 6.1
Topological space
¶
- depends_on equation 6.12 eq:top-euclidean-metric ¶
-
depends_on
definition 6.2
Open set
¶
- proves proof ch:11-manifolds-tensors-curvature@proof-22 ¶
Supports
- depends_on example 13.110 Closed but not exact: the angle form ¶
- depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
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 |
→ | Closed form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5181 |
depends_on |
→ | Exact form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5181 |
depends_on |
→ | Star-shaped domain | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5181 |
depends_on |
← | Closed but not exact: the angle form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5266 |
depends_on |
← | Symmetric analogue of the converse Poincaré lemma | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5306 |
proves |
← | ch:11-manifolds-tensors-curvature@proof-22 | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5184 |