Metric Tensor Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of the metric tensor within the Kireth Research Semantic Library. A metric tensor is a smoothly varying symmetric bilinear form on tangent spaces that determines geometric quantities such as lengths, angles, distances, volume elements, orthogonality and, through the Levi-Civita connection, geodesics and curvature. This ontology supports Riemannian and pseudo-Riemannian metrics, coordinate components, inverse metrics, line elements, raising and lowering indices, determinants, volume forms, induced metrics and relationships with manifolds, tangent spaces, connections, geodesics and curvature.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MetricTensor
 SYMBOLIC_FORMS = [
  "g",
  "g_ij",
  "ds^2=g_ij dx^i dx^j"
 ]
 ENTITY_CLASS = SymmetricCovariantTensorField
 PRIMARY_CONTEXT = DifferentialGeometry
 DEFINITION = "A metric tensor is a smoothly varying symmetric nondegenerate bilinear form on tangent spaces, positive definite in the Riemannian case."
 SEMANTIC_ROLE = "Provides the local measurement structure for length, angle, orthogonality, distance, volume, index movement, geodesics and curvature on smooth manifolds."
}

SCOPE {
 INCLUDE = [
  MetricTensor,
  RiemannianMetric,
  PseudoRiemannianMetric,
  MetricComponents,
  InverseMetric,
  BilinearForm,
  InnerProduct,
  LineElement,
  Norm,
  Length,
  Angle,
  Orthogonality,
  Distance,
  DeterminantMetric,
  VolumeElement,
  VolumeForm,
  MusicalIsomorphism,
  RaiseIndex,
  LowerIndex,
  InducedMetric,
  PullbackMetric,
  LeviCivitaConnection
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralDistanceFunctionWithoutTensorStructure,
  NonmetricConnection,
  ArbitraryBilinearFormWithoutGeometricMetricRole,
  DiscreteGraphMetricUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When metric occurs with tensor components, line elements, tangent vectors, inner products, index raising or lowering, geodesics or curvature, interpret it as a differential-geometric metric tensor."
}

CORE_CONCEPTS {

 MetricTensor {
  symbol = "g"
  definition = "A smooth symmetric nondegenerate covariant 2-tensor field."
 }

 RiemannianMetric {
  definition = "A metric tensor that is positive definite on every nonzero tangent vector."
 }

 PseudoRiemannianMetric {
  definition = "A nondegenerate symmetric metric tensor with indefinite signature."
 }

 MetricComponents {
  symbol = "g_ij"
  definition = "Coordinate components of the metric in a chosen basis."
 }

 InverseMetric {
  symbol = "g^ij"
  definition = "The inverse matrix of g_ij in coordinates."
  relation = "g^ik g_kj = delta^i_j"
 }

 LineElement {
  relation = "ds^2=g_ij dx^i dx^j"
 }

 Norm {
  definition = "For Riemannian metric g, |v|=sqrt(g(v,v))."
 }

 Angle {
  relation = "cos(theta)=g(u,v)/(|u||v|)"
  condition = "u!=0 AND v!=0 in positive-definite metric"
 }

 Orthogonality {
  relation = "g(u,v)=0"
 }

 VolumeElement {
  relation = "sqrt(det(g_ij)) dx^1...dx^n"
  condition = "Riemannian coordinates"
 }

 MusicalIsomorphism {
  definition = "Metric-induced identification between tangent and cotangent vectors."
 }

 InducedMetric {
  definition = "A metric on a submanifold obtained by restricting or pulling back an ambient metric."
 }
}

RELATIONS {
 MetricTensor --is_tensor_field_on--> SmoothManifold
 RiemannianMetric --is_special_case_of--> MetricTensor
 PseudoRiemannianMetric --is_special_case_of--> MetricTensor
 MetricTensor --acts_on--> TangentVector
 MetricTensor --defines--> Norm
 MetricTensor --defines--> Angle
 MetricTensor --defines--> Orthogonality
 MetricTensor --defines--> Length
 MetricTensor --induces--> Distance
 MetricTensor --determines--> VolumeElement
 MetricTensor --has_inverse--> InverseMetric
 MetricTensor --defines--> MusicalIsomorphism
 MetricTensor --determines--> LeviCivitaConnection WHEN Riemannian_or_PseudoRiemannian
 LeviCivitaConnection --determines--> Geodesic
 LeviCivitaConnection --determines--> Curvature
 Embedding --can_induce--> InducedMetric
 SmoothMap --can_pull_back--> MetricTensor
}

OPERATORS {
 METRIC_PAIRING(g,u,v) :=
  g(u,v)

 NORM(g,v) :=
  sqrt(g(v,v))
  WHEN g(v,v)>=0 in applicable context

 ANGLE(g,u,v) :=
  arccos(g(u,v)/(sqrt(g(u,u))*sqrt(g(v,v))))
  WHEN positive_definite AND u!=0 AND v!=0

 IS_ORTHOGONAL(g,u,v) :=
  g(u,v)=0

 LINE_ELEMENT(g,dx) :=
  g_ij dx^i dx^j

 INVERSE_METRIC(g_ij) :=
  matrix_inverse(g_ij)
  WHEN det(g_ij)!=0

 LOWER_INDEX(g,v) :=
  v_flat_i = g_ij v^j

 RAISE_INDEX(g_inverse,omega) :=
  omega_sharp^i = g^ij omega_j

 VOLUME_DENSITY(g) :=
  sqrt(abs(det(g_ij)))

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

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

 PULLBACK_METRIC(F,g) :=
  F^*g
}

DEFINITIONS {
 D1 = "A metric tensor is symmetric."
 D2 = "A metric tensor is nondegenerate."
 D3 = "A Riemannian metric is positive definite."
 D4 = "The inverse metric satisfies g^ik g_kj=delta^i_j."
 D5 = "The metric defines lengths and angles of tangent vectors."
 D6 = "The metric lowers vector indices and its inverse raises covector indices."
 D7 = "The determinant of the metric determines the local volume density."
 D8 = "The Levi-Civita connection is uniquely determined by the metric."
 D9 = "An induced metric is the pullback of an ambient metric along an embedding."
 D10 = "Metric components are coordinate dependent, while the metric tensor itself is geometric."
}

AXIOMS {
 A1_SYMMETRY:
  g(u,v)=g(v,u)

 A2_NONDEGENERACY:
  If g(u,v)=0 for all v then u=0

 A3_RIEMANNIAN_POSITIVITY:
  For Riemannian g, g(v,v)>0 for all nonzero v

 A4_INVERSE:
  g^ik g_kj=delta^i_j

 A5_SMOOTHNESS:
  Metric components vary smoothly in smooth coordinates

 A6_LENGTH:
  Curve length is integral sqrt(g(gamma_dot,gamma_dot)) dt

 A7_LEVI_CIVITA:
  A metric determines a unique torsion-free metric-compatible connection

 A8_COORDINATE_INVARIANCE:
  Geometric quantities derived from g are invariant under coordinate changes when transformed correctly
}

REASONING {

 R1_METRIC_VALIDITY {
  INPUT = candidate_tensor_g
  TEST = symmetry
  TEST = nondegeneracy
  TEST = smoothness
  IF Riemannian_context
   TEST = positive_definiteness
  RETURN = ValidMetricOrNot
 }

 R2_NORM_ANGLE {
  INPUT = [g,u,v]
  COMPUTE = g(u,u),g(v,v),g(u,v)
  IF norm_requested
   RETURN = sqrt(g(v,v))
  IF angle_requested AND positive_definite
   RETURN = arccos(g(u,v)/(sqrt(g(u,u))*sqrt(g(v,v))))
 }

 R3_INVERSE_METRIC {
  INPUT = g_ij
  COMPUTE = determinant
  REQUIRE = determinant!=0
  INVERT = matrix
  VERIFY = g^ik g_kj=delta^i_j
  RETURN = g^ij
 }

 R4_INDEX_MOVEMENT {
  INPUT = tensor_components_and_metric
  IF lower_index
   CONTRACT = with g_ij
  ELSE IF raise_index
   CONTRACT = with g^ij
  RETURN = transformed_components
 }

 R5_CURVE_LENGTH {
  INPUT = [g,gamma]
  COMPUTE = gamma_dot
  COMPUTE = speed=sqrt(g(gamma_dot,gamma_dot))
  INTEGRATE = speed
  RETURN = length
 }

 R6_VOLUME_ELEMENT {
  INPUT = coordinate_metric_matrix
  COMPUTE = determinant
  RETURN = sqrt(abs(det(g))) coordinate_volume_density
 }

 R7_LEVI_CIVITA {
  INPUT = g_ij
  COMPUTE = g^ij
  COMPUTE = Christoffel_symbols
  RETURN = LeviCivitaConnection
 }

 R8_INDUCED_METRIC {
  INPUT = [embedding_F,ambient_metric_g]
  COMPUTE = F^*g
  RETURN = metric_on_submanifold
 }

 R9_COORDINATE_TRANSFORMATION {
  INPUT = metric_components_and_coordinate_change
  APPLY = covariant_rank_two transformation law
  RETURN = transformed_metric_components
 }
}

CONSTRAINTS {
 C1 = "Metric tensor components depend on coordinate basis."
 C2 = "Riemannian and pseudo-Riemannian signatures must not be conflated."
 C3 = "Positive-definite angle formulas do not directly transfer to null or indefinite cases."
 C4 = "Inverse metric exists only for nondegenerate metrics."
 C5 = "Index raising and lowering must use the metric and its inverse consistently."
 C6 = "The determinant of g is coordinate dependent, but the full volume form is geometrically meaningful."
 C7 = "Curve distance is generally defined as an infimum of lengths, not simply a coordinate difference."
 C8 = "Metric compatibility belongs to the connection, not merely coordinate constancy of g_ij."
 C9 = "A conformally related metric generally changes lengths and geodesics."
}

SEMANTIC_EQUIVALENCE {
 "MetricTensor" <=> "SmoothSymmetricNondegenerateCovariantTwoTensor"
 "RiemannianMetric" <=> "PositiveDefiniteMetricTensor"
 "InverseMetric" <=> "ContravariantMetricTensor"
 "LineElement" <=> "MetricQuadraticFormOnCoordinateDifferentials"
 "LowerIndex" <=> "FlatMusicalMap"
 "RaiseIndex" <=> "SharpMusicalMap"
}

VALIDATION {
 V1 = "Metric is symmetric."
 V2 = "Metric is nondegenerate."
 V3 = "Riemannian metric is positive definite."
 V4 = "Inverse metric satisfies matrix inverse identity."
 V5 = "Norm and angle formulas use the correct metric signature."
 V6 = "Index movement contracts correct index positions."
 V7 = "Volume density uses determinant magnitude appropriately."
 V8 = "Levi-Civita construction uses metric and first derivatives consistently."
 V9 = "Coordinate transformation obeys covariant tensor law."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 METRIC_TENSOR_DEFINITION = 0.99
 VALIDITY_REASONING = 0.99
 NORM_ANGLE_REASONING = 0.99
 INVERSE_METRIC_REASONING = 0.99
 INDEX_MOVEMENT_REASONING = 0.99
 VOLUME_REASONING = 0.98
 LEVI_CIVITA_REASONING = 0.98
 INDUCED_METRIC_REASONING = 0.99

 POLICY =
  "Reduce confidence when metric signature, coordinate basis, index convention, manifold dimension, degeneracy or pullback map are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MetricTensorReasoning,
  DifferentialGeometryReasoning,
  RiemannianGeometryReasoning,
  NormReasoning,
  AngleReasoning,
  DistanceReasoning,
  IndexReasoning,
  VolumeReasoning,
  ConnectionReasoning,
  GeodesicReasoning,
  CurvatureReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "metric tensor" -> MetricTensor
  "Riemannian metric" -> RiemannianMetric
  "pseudo-Riemannian metric" -> PseudoRiemannianMetric
  "inverse metric" -> InverseMetric
  "metric components" -> MetricComponents
  "line element" -> LineElement
  "raise index" -> RaiseIndex
  "lower index" -> LowerIndex
  "volume element" -> VolumeElement
  "induced metric" -> InducedMetric
 }

 INTERPRETATION_ORDER = [
  detect_metric_tensor_context,
  identify manifold_dimension_coordinate_basis_and_signature,
  identify metric_components_and_invertibility,
  classify Riemannian_pseudoRiemannian_induced_or_pullback context,
  identify norm_angle_length_distance_volume_index_connection_or_curvature goal,
  perform metric_specific reasoning,
  verify symmetry_nondegeneracy_signature_and_coordinate constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("metric tensor")
  OR tensor_g_with_differential_geometry_context
  THEN activate(MetricTensorOntology)

 E2:
  IF metric_validity_requested
  THEN test symmetry_nondegeneracy_and_signature

 E3:
  IF inverse_metric_requested
  THEN require det(g)!=0
  AND invert coordinate_matrix

 E4:
  IF norm_or_angle_requested
  THEN evaluate metric_pairings
  AND respect signature constraints

 E5:
  IF index_raising_or_lowering_requested
  THEN contract with g^ij_or_g_ij appropriately

 E6:
  IF curve_length_requested
  THEN integrate metric_speed

 E7:
  IF volume_element_requested
  THEN use sqrt(abs(det(g)))

 E8:
  IF LeviCivita_connection_requested
  THEN compute Christoffel_symbols from g and g_inverse

 E9:
  IF induced_metric_requested
  THEN pull_back ambient_metric_by_embedding
}

RECURSION {
 ENTRY = MetricTensor

 EXPAND(MetricTensor) -> [
  RiemannianMetric,
  PseudoRiemannianMetric,
  MetricComponents,
  InverseMetric,
  LineElement,
  Norm,
  Angle,
  Orthogonality,
  Length,
  Distance,
  DeterminantMetric,
  VolumeElement,
  MusicalIsomorphism,
  InducedMetric,
  PullbackMetric,
  LeviCivitaConnection
 ]

 EXPAND(MusicalIsomorphism) -> [
  FlatMap,
  SharpMap,
  LowerIndex,
  RaiseIndex,
  TangentVector,
  CotangentVector
 ]

 EXPAND(RiemannianMetric) -> [
  PositiveDefiniteness,
  Length,
  Angle,
  Distance,
  VolumeForm,
  LeviCivitaConnection,
  Geodesic,
  Curvature
 ]

 EXPAND(LeviCivitaConnection) -> [
  ChristoffelSymbol,
  CovariantDerivative,
  Geodesic,
  RiemannCurvature
 ]

 STOP_CONDITION =
  "Stop when the metric's validity, signature, inverse, measurement structure, index operation, volume element, induced metric, connection or geodesic/curvature consequence is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Differential Geometry"

 IMMEDIATE_CONTEXTS = [
  "Differential Geometry Ontology",
  "Manifold Ontology",
  "Geodesic Ontology",
  "Curvature Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Differential Geometry",
  "Differential Geometry Ontology",
  "Manifold Ontology",
  "Curve Ontology",
  "Surface Ontology",
  "Geodesic Ontology",
  "Curvature Ontology",
  "Vector Geometry Ontology",
  "Linear Algebra Ontologies",
  "Tensor Ontologies",
  "Riemannian Geometry Ontologies"
 ]

 ROUTE_RULES = [
  "Metric Tensor -> Differential Geometry Ontology for parent smooth-geometry semantics",
  "Metric Tensor -> Manifold for tangent-space and coordinate context",
  "Metric Tensor -> Vector Geometry and Linear Algebra for bilinear-form operations",
  "Metric Tensor -> Tensor Ontologies for index and component transformations",
  "Metric Tensor -> Geodesic for length-stationary curves and distance",
  "Metric Tensor -> Curvature for Levi-Civita-derived curvature",
  "Metric Tensor -> Surface for induced first fundamental form",
  "Metric Tensor -> Riemannian Geometry for full metric-manifold reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible metric structures, signatures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve symmetric nondegenerate tensor semantics.",
  "Preserve Riemannian positivity distinction.",
  "Preserve coordinate-component versus tensor distinction.",
  "Preserve inverse-metric relation.",
  "Preserve raising and lowering semantics.",
  "Preserve metric-to-Levi-Civita relationship.",
  "Preserve volume and length constructions.",
  "Preserve machine-readable metric-tensor semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A metric tensor is a smooth symmetric nondegenerate covariant 2-tensor defining geometric measurement on tangent spaces."

 LINE_ELEMENT_SUMMARY =
  "ds^2=g_ij dx^i dx^j"

 INVERSE_SUMMARY =
  "g^ik g_kj=delta^i_j"

 NORM_SUMMARY =
  "|v|=sqrt(g(v,v)) in the Riemannian case."

 ANGLE_SUMMARY =
  "cos(theta)=g(u,v)/(|u||v|) for nonzero vectors in a positive-definite metric."

 LOWER_INDEX_SUMMARY =
  "v_flat_i=g_ij v^j"

 RAISE_INDEX_SUMMARY =
  "omega_sharp^i=g^ij omega_j"

 VOLUME_SUMMARY =
  "dV=sqrt(det(g_ij)) dx^1...dx^n for a Riemannian metric in coordinates."

 CONNECTION_SUMMARY =
  "The metric uniquely determines its Levi-Civita connection."
}

FLOW {
 INPUT
 -> detect_metric_tensor_context
 -> identify manifold_dimension_coordinates_signature_and_metric_components
 -> verify symmetry_nondegeneracy_and_positive_definiteness_when_Riemannian
 -> identify inverse_norm_angle_length_distance_volume_index_induced_metric_connection_or_curvature goal
 -> perform metric_specific tensor_and_measurement reasoning
 -> verify coordinate_basis_signature_index_and_determinant constraints
 -> route to differential_geometry manifold vector linear_algebra tensor geodesic curvature surface_or_riemannian ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Metric Tensor Ontology defines the smooth symmetric nondegenerate bilinear form that supplies measurement structure on manifolds. It supports Riemannian and pseudo-Riemannian signatures, coordinate components, inverse metrics, lengths, angles, orthogonality, curve length, distance, volume elements, index raising and lowering, induced metrics and the Levi-Civita connection linking the metric to geodesics and curvature."
}

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

}

END_FLOW