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gptkbp:instanceOf
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gptkb:mathematical_concept
gptkb:Riemannian_manifold
Kähler manifold
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gptkbp:alternativeName
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Lorentzian_manifold
Riemann_surface
Riemannian_metric
differentiable_manifold
manifold
smooth_manifold
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gptkbp:assignee
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inner product
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gptkbp:assignsTo
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tangent space
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gptkbp:atEachPoint
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gptkb:Riemannian_manifold
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gptkbp:automorphismGroup
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finite for compact surfaces of genus > 1
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gptkbp:canBe
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compact
connected
disconnected
non-compact
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gptkbp:category
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gptkb:algebra
gptkb:geometry
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gptkbp:class
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gptkb:genus
complex structure
topological type
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gptkbp:containsGenus
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gptkb:integral
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gptkbp:definedIn
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gptkb:Riemannian_manifold
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gptkbp:dimensions
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one
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gptkbp:enables
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distance measurement
definition of curvature
definition of geodesics
measurement of angles
measurement of lengths
measurement of volumes
angle measurement
volume measurement
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gptkbp:example
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gptkb:complex_plane
gptkb:butter
gptkb:Riemann_sphere
modular group
hyperelliptic curve
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gptkbp:field
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gptkb:algebraic_geometry
gptkb:mathematics
gptkb:topology
complex analysis
differential geometry
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gptkbp:generalizes
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gptkb:Euclidean_space
Euclidean metric
surface
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gptkbp:hasConcept
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gptkb:Einstein_field_equations
gptkb:Weyl_tensor
gptkb:geometry
gptkb:holonomy_group
gptkb:Hadamard_manifold
gptkb:Hodge_theory
gptkb:spectral_geometry
gptkb:sphere_theorem
gptkb:Riemann_curvature_tensor
gptkb:Lie_group
gptkb:Atiyah-Singer_index_theorem
gptkb:Bochner_technique
gptkb:Gauss-Bonnet_theorem
gptkb:Hopf-Rinow_theorem
gptkb:Laplace-Beltrami_operator
gptkb:Myers'_theorem
gptkb:Riemannian_metric_tensor
gptkb:Riemannian_orbifold
gptkb:Cartan's_theorem
gptkb:Ricci_flow
gptkb:de_Rham_cohomology
gptkb:Bonnet-Myers_theorem
gptkb:Cartan-Hadamard_theorem
gptkb:Jacobi_field
gptkb:heat_kernel
gptkb:isometry_group
gptkb:spin_manifold
completeness
exponential map
Kähler manifold
distance function
isometry
orthogonal group
harmonic forms
holonomy
homogeneous space
symmetric space
Dirac operator
Einstein manifold
Killing vector field
Levi-Civita connection
Ricci curvature
Riemannian covering
Riemannian foliation
Riemannian submersion
Riemannian symmetric space
curvature
cut locus
frame bundle
harmonic map
injectivity radius
minimal surface
parallel transport
scalar curvature
second fundamental form
sectional curvature
submanifold
volume form
cotangent bundle
tangent bundle
geodesic completeness
normal coordinates
comparison theorem
conjugate points
index theorem
metric connection
orthonormal frame
parallel vector field
symmetric tensor
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gptkbp:hasProperty
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gptkb:Hausdorff
metric tensor
second-countable
connected (if not specified otherwise)
locally homeomorphic to open subset of complex plane
smoothness
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gptkbp:hasSpecialCase
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gptkb:pseudo-Riemannian_metric
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gptkbp:heldBy
|
positive-definite symmetric bilinear form
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|
https://www.w3.org/2000/01/rdf-schema#label
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Riemannian manifold
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gptkbp:introduced
|
gptkb:Bernhard_Riemann
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gptkbp:introducedIn
|
1854
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gptkbp:namedAfter
|
gptkb:Bernhard_Riemann
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gptkbp:notation
|
g
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gptkbp:relatedTo
|
gptkb:fundamental_group
gptkb:algebraic_geometry
gptkb:Abelian_differential
gptkb:Jacobian_variety
gptkb:Kleinian_group
gptkb:Schottky_group
gptkb:Weierstrass_point
gptkb:uniformization_theorem
analytic continuation
distance function
divisor
metric tensor
modular group
moduli space
sheaf
covering space
mapping class group
branched covering
Levi-Civita connection
Ricci curvature
holomorphic function
meromorphic function
monodromy
period matrix
scalar curvature
sectional curvature
theta function
universal covering space
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gptkbp:structure
|
gptkb:Riemannian_manifold
conformal structure
complex structure
|
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gptkbp:studiedBy
|
gptkb:Bernhard_Riemann
|
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gptkbp:studiedIn
|
gptkb:Teichmüller_theory
gptkb:moduli_space_theory
|
|
gptkbp:universalCover
|
gptkb:complex_plane
gptkb:Riemann_sphere
gptkb:unit_disk
|
|
gptkbp:usedIn
|
gptkb:general_relativity
gptkb:topology
gptkb:Riemannian_geometry
gptkb:string_theory
mathematical physics
global analysis
|
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gptkbp:bfsParent
|
gptkb:geometry
|
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gptkbp:bfsLayer
|
4
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