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gptkbp:instanceOf
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gptkb:mathematical_concept
gptkb:topology
gptkb:Riemannian_manifold
gptkb:Lie_group
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|
gptkbp:canBeDefinedAs
|
quotient group R/Z
set of points in R^2 at distance 1 from origin
unit complex numbers
|
|
gptkbp:compact
|
true
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gptkbp:dimensions
|
1
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gptkbp:Euler_characteristic
|
0
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gptkbp:fundamentalGroup
|
Z
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gptkbp:hasConnection
|
false
true
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|
gptkbp:hasGroupStructure
|
true
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gptkbp:homologyGroup
|
H_0 = Z
H_1 = Z
H_n = 0 for n > 1
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gptkbp:homotopyType
|
gptkb:butter
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
S^1
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gptkbp:isBoundaryOf
|
disk D^2
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|
gptkbp:isClosedManifold
|
true
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|
gptkbp:isCompactLieGroup
|
true
|
|
gptkbp:isComplexManifold
|
false
|
|
gptkbp:isConnectedLieGroup
|
true
|
|
gptkbp:isFundamentalGroupOf
|
gptkb:butter
|
|
gptkbp:isHomogeneousSpace
|
true
|
|
gptkbp:isHomotopyEquivalentTo
|
gptkb:butter
|
|
gptkbp:isKählerManifold
|
false
|
|
gptkbp:isNonAbelian
|
true
|
|
gptkbp:isomorphicTo
|
gptkb:butter
|
|
gptkbp:isOneDimensionalLieGroup
|
true
|
|
gptkbp:isOrientable
|
true
|
|
gptkbp:isParallelizable
|
true
|
|
gptkbp:isPathConnected
|
true
|
|
gptkbp:isRealManifold
|
true
|
|
gptkbp:isRiemannianManifold
|
true
|
|
gptkbp:isSimpleConnected
|
false
|
|
gptkbp:isSmoothManifold
|
true
|
|
gptkbp:isSpinManifold
|
true
|
|
gptkbp:isSymmetricSpace
|
true
|
|
gptkbp:relatedGroup
|
multiplication of complex numbers
addition modulo 1
|
|
gptkbp:subclassOf
|
R^2
|
|
gptkbp:universalCover
|
real line
itself
|
|
gptkbp:usedIn
|
gptkb:topology
Fourier analysis
complex analysis
differential geometry
homotopy theory
|
|
gptkbp:bfsParent
|
gptkb:unit_circle_in_complex_plane
|
|
gptkbp:bfsLayer
|
7
|