Differential Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the core Differential Geometry Ontology within the Kireth Research Semantic Library. Differential geometry studies smooth curves, surfaces and manifolds using differential calculus, linear algebra, tensor methods and geometric analysis. This ontology acts as the central semantic node beneath the Differential Geometry parent layer and routes into tangent spaces, metrics, connections, geodesics, curvature, differential forms, embeddings, immersions and diffeomorphisms.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Differential Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Core Differential Geometry / Smooth Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/differential-geometry/differential-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = DifferentialGeometryOntology
 ENTITY_CLASS = SmoothGeometryFramework
 PRIMARY_CONTEXT = DifferentialGeometry
 DEFINITION = "Differential geometry is the branch of geometry concerned with smooth curves, surfaces and manifolds together with differentiable structures, metrics, connections, geodesics and curvature."
 SEMANTIC_ROLE = "Provides the core semantic framework linking smooth manifolds, tangent spaces, differential forms, metrics, covariant differentiation, geodesics and curvature."
}

SCOPE {
 INCLUDE = [
  SmoothCurve,
  SmoothSurface,
  SmoothManifold,
  CoordinateChart,
  Atlas,
  TangentSpace,
  TangentVector,
  CotangentSpace,
  VectorField,
  DifferentialForm,
  TensorField,
  RiemannianMetric,
  Connection,
  LeviCivitaConnection,
  CovariantDerivative,
  ParallelTransport,
  Geodesic,
  RiemannCurvature,
  RicciCurvature,
  ScalarCurvature,
  GaussianCurvature,
  MeanCurvature,
  FirstFundamentalForm,
  SecondFundamentalForm,
  ShapeOperator,
  Embedding,
  Immersion,
  Diffeomorphism
 ]

 EXCLUDE_AS_PRIMARY = [
  PurelyTopologicalStructureWithoutSmoothData,
  DiscreteGeometryWithoutDifferentialStructure,
  AlgebraicGeometryWithoutDifferentialContext,
  PhysicalFieldTheoryWithoutMathematicalGeometricAbstraction
 ]

 DISAMBIGUATION_RULE =
  "Interpret the context as differential geometry when smoothness, tangent structure, metrics, geodesics, curvature, differential forms or differentiable maps are central."
}

CORE_CONCEPTS {

 SmoothManifold {
  definition = "A topological manifold equipped with a maximal smooth atlas."
 }

 CoordinateChart {
  definition = "A map from an open subset of a manifold to an open subset of Euclidean space."
 }

 TangentSpace {
  symbol = "T_pM"
  definition = "The vector space of tangent directions to manifold M at point p."
 }

 CotangentSpace {
  symbol = "T_p^*M"
  definition = "The dual vector space of T_pM."
 }

 VectorField {
  definition = "A smooth section of the tangent bundle."
 }

 DifferentialForm {
  definition = "A smooth alternating covariant tensor field."
 }

 TensorField {
  definition = "A smoothly varying multilinear object defined over a manifold."
 }

 RiemannianMetric {
  symbol = "g"
  definition = "A smoothly varying positive-definite inner product on tangent spaces."
 }

 Connection {
  definition = "A rule for differentiating vector fields in a geometrically meaningful way."
 }

 LeviCivitaConnection {
  definition = "The unique torsion-free connection compatible with a Riemannian metric."
 }

 Geodesic {
  definition = "A curve satisfying zero covariant acceleration."
 }

 RiemannCurvature {
  definition = "The curvature tensor measuring noncommutativity of covariant differentiation."
 }

 RicciCurvature {
  definition = "A contraction of the Riemann curvature tensor."
 }

 ScalarCurvature {
  definition = "The trace of the Ricci tensor with respect to the metric."
 }

 GaussianCurvature {
  definition = "For a surface, the product of principal curvatures."
 }

 Embedding {
  definition = "A smooth immersion that is a homeomorphism onto its image."
 }

 Immersion {
  definition = "A smooth map whose differential is injective at every point."
 }

 Diffeomorphism {
  definition = "A bijective smooth map with smooth inverse."
 }
}

RELATIONS {
 DifferentialGeometryOntology --is_child_of--> DifferentialGeometry
 SmoothManifold --has--> CoordinateChart
 SmoothManifold --has--> Atlas
 SmoothManifold --has_at_point--> TangentSpace
 TangentSpace --has_dual--> CotangentSpace
 TangentBundle --contains_sections--> VectorField
 CotangentBundle --supports--> DifferentialForm
 SmoothManifold --can_carry--> RiemannianMetric
 RiemannianMetric --determines--> LeviCivitaConnection
 LeviCivitaConnection --defines--> Geodesic
 Connection --defines--> ParallelTransport
 Connection --defines--> Curvature
 RiemannCurvature --contracts_to--> RicciCurvature
 RicciCurvature --contracts_to--> ScalarCurvature
 EmbeddedSurface --has--> FirstFundamentalForm
 EmbeddedSurface --has--> SecondFundamentalForm
 SecondFundamentalForm --relates_to--> ShapeOperator
 ShapeOperator --determines--> PrincipalCurvatures
 PrincipalCurvatures --determine--> GaussianCurvature
}

OPERATORS {
 TANGENT_SPACE(M,p) :=
  T_pM

 COTANGENT_SPACE(M,p) :=
  T_p^*M

 DIFFERENTIAL(F,p) :=
  dF_p

 PUSHFORWARD(F,p,v) :=
  dF_p(v)

 PULLBACK(F,omega) :=
  F^*(omega)

 COVARIANT_DERIVATIVE(X,Y) :=
  nabla_X(Y)

 PARALLEL_TRANSPORT(gamma,v0) :=
  solve nabla_gamma_dot(V)=0 with V(0)=v0

 GEODESIC_EQUATION(local_coordinates,Gamma) :=
  d2x^k/dt^2 + Gamma^k_ij*(dx^i/dt)*(dx^j/dt)=0

 RIEMANN_CURVATURE(X,Y,Z) :=
  nabla_X(nabla_Y Z)-nabla_Y(nabla_X Z)-nabla_[X,Y]Z

 RICCI(R) :=
  trace_of_Riemann_curvature

 SCALAR_CURVATURE(g,Ric) :=
  trace_g(Ric)

 LENGTH(gamma,g) :=
  integral sqrt(g(gamma_dot,gamma_dot)) dt

 ENERGY(gamma,g) :=
  (1/2)*integral g(gamma_dot,gamma_dot) dt

 GAUSSIAN_CURVATURE(k1,k2) :=
  k1*k2

 MEAN_CURVATURE(k1,k2) :=
  (k1+k2)/2
}

DEFINITIONS {
 D1 = "Smooth manifolds are locally Euclidean spaces with compatible smooth charts."
 D2 = "Tangent spaces vary with the point."
 D3 = "Cotangent spaces are dual to tangent spaces."
 D4 = "A Riemannian metric defines lengths and angles on tangent spaces."
 D5 = "The Levi-Civita connection is torsion-free and metric-compatible."
 D6 = "Geodesics have zero covariant acceleration."
 D7 = "Curvature measures failure of local flatness."
 D8 = "Differential forms naturally pull back under smooth maps."
 D9 = "Immersions have injective differential."
 D10 = "Diffeomorphisms preserve smooth structure."
}

AXIOMS {
 A1_SMOOTH_ATLAS:
  Smooth-chart transition maps are smooth

 A2_TANGENT_LINEARITY:
  T_pM is a vector space

 A3_METRIC:
  A Riemannian metric is symmetric positive definite and smooth

 A4_CONNECTION:
  A connection satisfies linearity and Leibniz properties

 A5_LEVI_CIVITA:
  Every Riemannian metric has a unique torsion-free metric-compatible connection

 A6_GEODESIC:
  nabla_gamma_dot(gamma_dot)=0

 A7_CURVATURE:
  R(X,Y)Z is tensorial in X,Y,Z

 A8_COORDINATE_INDEPENDENCE:
  Geometric meaning is invariant under smooth coordinate changes
}

REASONING {

 R1_SMOOTH_MANIFOLD {
  INPUT = candidate_space
  TEST = manifold_axioms
  IDENTIFY = coordinate_charts
  VERIFY = smooth_transition_maps
  RETURN = SmoothManifoldOrNot
 }

 R2_TANGENT_SPACE {
  INPUT = [M,p]
  CONSTRUCT = tangent_vectors_as_curve_velocities_or_derivations
  VERIFY = vector_space_structure
  RETURN = T_pM
 }

 R3_SMOOTH_MAP_DIFFERENTIAL {
  INPUT = [F:M->N,p]
  COMPUTE = coordinate_Jacobian
  INTERPRET = linear_map dF_p:T_pM->T_F(p)N
  RETURN = differential
 }

 R4_METRIC {
  INPUT = [g,p,v,w]
  COMPUTE = g_p(v,w)
  DERIVE = norm_angle_length_or_volume_element
  RETURN = metric_result
 }

 R5_CONNECTION {
  INPUT = [nabla,X,Y]
  COMPUTE = nabla_X(Y)
  CHECK = connection_axioms
  RETURN = covariant_derivative
 }

 R6_GEODESIC {
  INPUT = metric_or_connection_and_initial_data
  COMPUTE = Christoffel_symbols_when_coordinates_used
  SOLVE = geodesic_equation
  RETURN = geodesic
 }

 R7_CURVATURE {
  INPUT = metric_or_connection
  COMPUTE = Riemann_curvature
  CONTRACT = Ricci_and_scalar_when_requested
  RETURN = curvature_result
 }

 R8_SURFACE_GEOMETRY {
  INPUT = embedded_surface
  COMPUTE = first_fundamental_form
  COMPUTE = second_fundamental_form
  DERIVE = shape_operator
  DERIVE = principal_curvatures
  RETURN = Gaussian_mean_and_extrinsic_geometry
 }

 R9_DIFFERENTIAL_FORM {
  INPUT = form_and_operation
  IF exterior_derivative_requested
   COMPUTE = d(omega)
  ELSE IF pullback_requested
   COMPUTE = F^*(omega)
  ELSE IF integration_context
   VERIFY = orientation_and_degree
  RETURN = form_result
 }

 R10_MAP_CLASSIFICATION {
  INPUT = smooth_map F
  TEST = differential_rank
  IF injective_differential_everywhere
   CLASSIFY = Immersion
  IF immersion_and_homeomorphism_onto_image
   CLASSIFY = Embedding
  IF bijective_smooth_with_smooth_inverse
   CLASSIFY = Diffeomorphism
  RETURN = map_classification
 }
}

CONSTRAINTS {
 C1 = "Coordinate representations must be distinguished from invariant geometric objects."
 C2 = "Tangent vectors at different points belong to different vector spaces."
 C3 = "Christoffel symbols are not tensor components."
 C4 = "Riemannian and pseudo-Riemannian metrics must not be conflated."
 C5 = "Curvature sign conventions may differ."
 C6 = "Intrinsic and extrinsic curvature must remain distinct."
 C7 = "Geodesics need not be globally distance-minimising."
 C8 = "A smooth immersion need not be an embedding."
 C9 = "A bijective smooth map need not be a diffeomorphism unless the inverse is smooth."
 C10 = "Pullback and pushforward act on different geometric object types."
}

SEMANTIC_EQUIVALENCE {
 "DifferentialGeometryOntology" <=> "CoreSmoothGeometryOntology"
 "SmoothManifold" <=> "DifferentiableManifold"
 "TangentSpace" <=> "InfinitesimalDirectionSpaceAtPoint"
 "RiemannianMetric" <=> "SmoothPositiveDefiniteMetricTensor"
 "LeviCivitaConnection" <=> "UniqueTorsionFreeMetricCompatibleConnection"
 "Geodesic" <=> "ZeroCovariantAccelerationCurve"
 "Diffeomorphism" <=> "SmoothBijectionWithSmoothInverse"
}

VALIDATION {
 V1 = "Transition maps are smooth."
 V2 = "Tangent and cotangent objects are attached to the correct point."
 V3 = "Metric is symmetric and positive definite."
 V4 = "Levi-Civita connection is torsion-free and metric-compatible."
 V5 = "Geodesic equation matches the chosen connection."
 V6 = "Curvature conventions are internally consistent."
 V7 = "Tensorial claims are coordinate-independent."
 V8 = "Map classifications satisfy their differential and topological conditions."
 V9 = "Intrinsic and extrinsic quantities are not conflated."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DIFFERENTIAL_GEOMETRY_ONTOLOGY = 0.99
 MANIFOLD_REASONING = 0.99
 TANGENT_REASONING = 0.99
 METRIC_REASONING = 0.99
 CONNECTION_REASONING = 0.99
 GEODESIC_REASONING = 0.98
 CURVATURE_REASONING = 0.98
 MAP_CLASSIFICATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when smoothness class, coordinate convention, metric signature, curvature sign convention, manifold dimension, connection or map regularity are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  DifferentialGeometryReasoning,
  SmoothManifoldReasoning,
  TangentSpaceReasoning,
  MetricReasoning,
  ConnectionReasoning,
  GeodesicReasoning,
  CurvatureReasoning,
  DifferentialFormReasoning,
  SurfaceGeometryReasoning,
  MapClassificationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "differential geometry ontology" -> DifferentialGeometryOntology
  "differential geometry" -> DifferentialGeometryOntology WHEN ontology_context
  "smooth manifold" -> SmoothManifold
  "tangent space" -> TangentSpace
  "cotangent space" -> CotangentSpace
  "Riemannian metric" -> RiemannianMetric
  "Levi-Civita connection" -> LeviCivitaConnection
  "geodesic" -> Geodesic
  "Riemann curvature" -> RiemannCurvature
  "Ricci curvature" -> RicciCurvature
  "scalar curvature" -> ScalarCurvature
 }

 INTERPRETATION_ORDER = [
  detect_differential_geometry_context,
  identify smooth_space_curve_surface_map_or_tensor,
  identify point_dimension_coordinates_and_smoothness_when_needed,
  identify tangent_metric_connection_form_geodesic_curvature_or_map_classification goal,
  select invariant_or_coordinate representation,
  perform differential_geometric reasoning,
  verify smoothness_tensoriality_point_dependence_and_convention constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("differential geometry ontology")
  OR ontology_context_with_smooth_geometry
  THEN activate(DifferentialGeometryOntology)

 E2:
  IF tangent_space_requested
  THEN identify manifold_and_point
  AND return T_pM

 E3:
  IF differential_of_smooth_map_requested
  THEN compute coordinate_Jacobian
  AND interpret invariantly as dF_p

 E4:
  IF Riemannian_metric_context
  THEN route to length_angle_volume_and_LeviCivita reasoning

 E5:
  IF geodesic_context
  THEN use connection_or_metric
  AND solve zero_covariant_acceleration equation

 E6:
  IF curvature_context
  THEN identify requested curvature_type_and_sign_convention

 E7:
  IF surface_extrinsic_geometry_context
  THEN compute fundamental_forms_shape_operator_and_principal_curvatures

 E8:
  IF map_classification_context
  THEN distinguish immersion_embedding_and_diffeomorphism

 E9:
  IF differential_form_context
  THEN distinguish pullback_exterior_derivative_wedge_and_integration operations
}

RECURSION {
 ENTRY = DifferentialGeometryOntology

 EXPAND(DifferentialGeometryOntology) -> [
  SmoothCurve,
  SmoothSurface,
  SmoothManifold,
  CoordinateChart,
  Atlas,
  TangentSpace,
  CotangentSpace,
  VectorField,
  DifferentialForm,
  TensorField,
  RiemannianMetric,
  Connection,
  LeviCivitaConnection,
  CovariantDerivative,
  Geodesic,
  Curvature,
  Embedding,
  Immersion,
  Diffeomorphism
 ]

 EXPAND(SmoothManifold) -> [
  CoordinateChart,
  Atlas,
  TangentBundle,
  CotangentBundle,
  TensorBundle,
  DifferentialForm
 ]

 EXPAND(RiemannianMetric) -> [
  Norm,
  Angle,
  Length,
  VolumeForm,
  LeviCivitaConnection,
  Geodesic
 ]

 EXPAND(Curvature) -> [
  RiemannCurvature,
  RicciCurvature,
  ScalarCurvature,
  SectionalCurvature,
  GaussianCurvature
 ]

 EXPAND(SurfaceGeometry) -> [
  FirstFundamentalForm,
  SecondFundamentalForm,
  ShapeOperator,
  PrincipalCurvature,
  GaussianCurvature,
  MeanCurvature
 ]

 STOP_CONDITION =
  "Stop when the requested smooth manifold, tangent, metric, connection, geodesic, curvature, form, surface or smooth-map property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Differential Geometry"

 IMMEDIATE_CONTEXTS = [
  "Differential Geometry",
  "Geometry Ontology",
  "Coordinate / Analytic Geometry",
  "Vector Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Differential Geometry",
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Vector Geometry Ontology",
  "Linear Algebra Ontologies",
  "Tensor Ontologies",
  "Topology Ontologies",
  "Calculus & Analysis Ontologies",
  "Transformation Ontology",
  "Manifold Ontologies"
 ]

 ROUTE_RULES = [
  "Differential Geometry Ontology -> Differential Geometry for parent smooth-geometry framework",
  "Differential Geometry Ontology -> Coordinate Geometry for local charts",
  "Differential Geometry Ontology -> Vector Geometry for tangent vectors and fields",
  "Differential Geometry Ontology -> Linear Algebra for tangent spaces and tensor operations",
  "Differential Geometry Ontology -> Calculus for derivatives, differential equations and variation",
  "Differential Geometry Ontology -> Topology for manifold structure",
  "Differential Geometry Ontology -> Tensor Ontologies for metric and curvature tensors",
  "Differential Geometry Ontology -> Transformation for diffeomorphisms and smooth maps"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible smooth structures, curvature notions, tensor relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve smooth-manifold semantics.",
  "Preserve tangent-space point dependence.",
  "Preserve metric and connection distinction.",
  "Preserve coordinate-independent interpretation.",
  "Preserve intrinsic-versus-extrinsic distinction.",
  "Preserve curvature-convention awareness.",
  "Preserve immersion-embedding-diffeomorphism distinctions.",
  "Preserve machine-readable differential-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Differential geometry is the branch of geometry concerned with smooth curves, surfaces and manifolds and the differential structures defined on them."

 MANIFOLD_SUMMARY =
  "A smooth manifold is locally Euclidean and equipped with a compatible smooth atlas."

 TANGENT_SUMMARY =
  "T_pM is the tangent vector space at point p."

 METRIC_SUMMARY =
  "A Riemannian metric is a smooth positive-definite inner product on tangent spaces."

 CONNECTION_SUMMARY =
  "The Levi-Civita connection is the unique torsion-free metric-compatible connection."

 GEODESIC_SUMMARY =
  "Geodesics satisfy nabla_gamma_dot(gamma_dot)=0."

 CURVATURE_SUMMARY =
  "The Riemann tensor measures curvature; contractions produce Ricci and scalar curvature."

 MAP_SUMMARY =
  "Immersions have injective differentials, embeddings are immersed homeomorphic images, and diffeomorphisms are smooth bijections with smooth inverses."
}

FLOW {
 INPUT
 -> detect_differential_geometry_ontology_context
 -> identify smooth_manifold_curve_surface_map_form_tensor_or_metric_structure
 -> identify point_dimension_coordinate_chart_and_smoothness_when_needed
 -> identify tangent_metric_connection_geodesic_curvature_surface_form_or_map goal
 -> select invariant_or_coordinate representation
 -> perform differential_geometry_ontology reasoning
 -> verify smoothness_point_dependence_tensoriality_coordinate_and_convention constraints
 -> route to differential_geometry coordinate vector linear_algebra calculus topology tensor_or_transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Differential Geometry Ontology provides the core semantic model for smooth geometric reasoning. It connects smooth manifolds, tangent and cotangent spaces, vector fields, differential forms, Riemannian metrics, connections, geodesics, intrinsic and extrinsic curvature, embeddings, immersions and diffeomorphisms while maintaining coordinate-independent and convention-aware semantics."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-DIFFERENTIAL-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW