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gptkbp:instanceOf
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gptkb:mathematical_concept
gptkb:topology
gptkb:Lie_group
flat manifold
parallelizable manifold
commutative Lie group
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gptkbp:compact
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true
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gptkbp:definedIn
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Cartesian product of n circles
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gptkbp:dimensions
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n
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gptkbp:fundamentalGroup
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Z^n
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gptkbp:hasConnection
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true
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gptkbp:hasFlatMetric
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true
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gptkbp:hasSpecialCase
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1-torus is the circle S^1
2-torus is the torus T^2
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gptkbp:homologyGroup
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H_k(T^n) = (Z^{n \\choose k})
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https://www.w3.org/2000/01/rdf-schema#label
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n-torus T^n
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gptkbp:isAbelianLieGroup
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true
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gptkbp:isCompactConnectedLieGroup
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true
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gptkbp:isCompactLieGroup
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true
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gptkbp:isComplexManifold
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true
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gptkbp:isFiberBundleOver
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T^{n-1}
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gptkbp:isFlatManifold
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true
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gptkbp:isGroupManifold
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true
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gptkbp:isHomogeneousSpace
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true
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gptkbp:isHomotopyEquivalentTo
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product of n circles
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gptkbp:isKählerManifold
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true
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gptkbp:isMaximalTorusOf
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gptkb:Sp(n)
gptkb:SU(n)
gptkb:SO(2n)
itself
U(n)
other compact Lie groups
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gptkbp:isNonAbelian
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true
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gptkbp:isOrientable
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true
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gptkbp:isParallelizable
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true
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gptkbp:isParallelizableManifold
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true
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gptkbp:isPrincipalBundle
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true
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gptkbp:isProductOfCircles
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true
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gptkbp:isQuotientOf
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gptkb:R^n_/_Z^n
R^n by Z^n
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gptkbp:isRealManifold
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true
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gptkbp:isSmoothManifold
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true
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gptkbp:isSymplecticManifold
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true
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gptkbp:notation
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(S^1)^n
T^n
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gptkbp:universalCover
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R^n
itself
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gptkbp:bfsParent
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gptkb:torus_group
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gptkbp:bfsLayer
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5
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