Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:canBeEmbeddedIn |
Euclidean space R^N
|
| gptkbp:definedIn |
set of points in R^N at unit distance from the origin
|
| gptkbp:dimensions |
gptkb:N-1
|
| gptkbp:example |
S^1 is the circle
S^2 is the 2-sphere S^3 is the 3-sphere |
| gptkbp:fundamentalGroup |
Z_2 for N=2
trivial for N>2 |
| gptkbp:groupOfSymmetries |
orthogonal group O(N)
|
| gptkbp:hasConnection |
true for N>2
|
| gptkbp:homologyGroup |
H_k(S^{N-1})
|
| gptkbp:homotopyGroup |
pi_k(S^{N-1})
|
| https://www.w3.org/2000/01/rdf-schema#label |
sphere S^{N-1}
|
| gptkbp:isBoundaryOf |
N-dimensional ball
|
| gptkbp:isHomogeneousSpace |
O(N)/O(N-1)
|
| gptkbp:isOrientable |
true
|
| gptkbp:surface |
2π^{N/2}/Γ(N/2)
|
| gptkbp:topologicalType |
compact manifold
connected manifold |
| gptkbp:volumeEnclosed |
π^{N/2}/Γ(N/2+1)
|
| gptkbp:bfsParent |
gptkb:SO(N)
|
| gptkbp:bfsLayer |
6
|