cognitive/knowledge_base/mathematics/differential_geometry.md

Differential Geometry in Cognitive Modeling

---

type: mathematical_concept id: differential_geometry_001 created: 2024-02-06 modified: 2024-02-06 tags: [mathematics, differential-geometry, manifolds, connections] aliases: [riemannian-geometry, geometric-mechanics] semantic_relations:

• type: implements links:
  • ../../docs/research/research_documentation_index
  • information_geometry
• type: uses links:
  • tensor_calculus
  • lie_theory
• type: documented_by links:
  • ../../docs/guides/implementation_guides_index
  • ../../docs/api/api_documentation_index

---

Overview

Differential geometry provides the mathematical foundation for understanding the geometric structure of state spaces and belief manifolds in cognitive modeling. This document explores differential geometric concepts and their applications in active inference.

Manifold Theory

Differentiable Manifolds

class DifferentiableManifold:
    """
    Differentiable manifold implementation.
    
    Theory:
        - [[manifold_theory]]
        - [[differential_topology]]
        - [[smooth_structures]]
    Mathematics:
        - [[topology]]
        - [[calculus_on_manifolds]]
    """
    def __init__(self,
                 dimension: int,
                 atlas: Dict[str, Chart]):
        self.dim = dimension
        self.atlas = atlas
        self._validate_smooth_structure()
        
    def coordinate_change(self,
                        chart1: str,
                        chart2: str,
                        point: np.ndarray) -> np.ndarray:
        """Change coordinates between charts."""
        if not self._charts_overlap(chart1, chart2):
            raise ValueError("Charts do not overlap")
        return self._compute_transition(chart1, chart2, point)
    
    def tangent_space(self,
                     point: np.ndarray,
                     chart: str) -> TangentSpace:
        """Get tangent space at point."""
        return self._construct_tangent_space(point, chart)

Riemannian Metrics

class RiemannianMetric:
    """
    Riemannian metric implementation.
    
    Theory:
        - [[riemannian_geometry]]
        - [[metric_tensor]]
        - [[inner_product]]
    Mathematics:
        - [[differential_geometry]]
        - [[tensor_calculus]]
    """
    def __init__(self,
                 manifold: DifferentiableManifold):
        self.manifold = manifold
        
    def metric_tensor(self,
                     point: np.ndarray,
                     chart: str) -> np.ndarray:
        """Compute metric tensor at point."""
        # Get coordinate basis
        basis = self._coordinate_basis(point, chart)
        
        # Compute components
        g = self._compute_metric_components(basis)
        
        return g
    
    def distance(self,
                p: np.ndarray,
                q: np.ndarray,
                chart: str) -> float:
        """Compute Riemannian distance."""
        # Find geodesic
        gamma = self._solve_geodesic_equation(p, q)
        
        # Compute length
        return self._compute_curve_length(gamma)

Connections and Transport

Levi-Civita Connection

class LeviCivitaConnection:
    """
    Levi-Civita connection implementation.
    
    Theory:
        - [[riemannian_connection]]
        - [[parallel_transport]]
        - [[geodesics]]
    Mathematics:
        - [[differential_geometry]]
        - [[tensor_calculus]]
    """
    def __init__(self,
                 metric: RiemannianMetric):
        self.metric = metric
        
    def christoffel_symbols(self,
                          point: np.ndarray,
                          chart: str) -> np.ndarray:
        """Compute Christoffel symbols."""
        # Metric and derivatives
        g = self.metric.metric_tensor(point, chart)
        dg = self._metric_derivatives(point, chart)
        
        # Compute symbols
        gamma = self._compute_christoffel(g, dg)
        
        return gamma
    
    def parallel_transport(self,
                         vector: np.ndarray,
                         curve: Curve) -> np.ndarray:
        """Parallel transport vector along curve."""
        return self._solve_parallel_transport(vector, curve)

Geodesic Flow

class GeodesicFlow:
    """
    Geodesic flow implementation.
    
    Theory:
        - [[geodesic_equation]]
        - [[exponential_map]]
        - [[hamiltonian_flow]]
    Mathematics:
        - [[differential_geometry]]
        - [[symplectic_geometry]]
    """
    def __init__(self,
                 connection: LeviCivitaConnection):
        self.connection = connection
        
    def geodesic(self,
                initial_point: np.ndarray,
                initial_velocity: np.ndarray,
                time: float) -> np.ndarray:
        """Compute geodesic flow."""
        # Geodesic equation
        def geodesic_equation(t, state):
            x, v = state[:self.dim], state[self.dim:]
            gamma = self.connection.christoffel_symbols(x)
            return np.concatenate([v, -gamma.dot(v).dot(v)])
        
        # Solve ODE
        solution = solve_ivp(
            geodesic_equation,
            (0, time),
            np.concatenate([initial_point, initial_velocity])
        )
        
        return solution.y[:self.dim, -1]

Curvature Theory

Riemann Curvature

class RiemannCurvature:
    """
    Riemann curvature implementation.
    
    Theory:
        - [[curvature_tensor]]
        - [[sectional_curvature]]
        - [[ricci_curvature]]
    Mathematics:
        - [[differential_geometry]]
        - [[tensor_calculus]]
    """
    def __init__(self,
                 connection: LeviCivitaConnection):
        self.connection = connection
        
    def curvature_tensor(self,
                        point: np.ndarray,
                        chart: str) -> np.ndarray:
        """Compute Riemann curvature tensor."""
        # Connection coefficients
        gamma = self.connection.christoffel_symbols(point, chart)
        
        # Compute components
        R = self._compute_riemann_components(gamma)
        
        return R
    
    def sectional_curvature(self,
                           point: np.ndarray,
                           plane: np.ndarray,
                           chart: str) -> float:
        """Compute sectional curvature."""
        # Curvature tensor
        R = self.curvature_tensor(point, chart)
        
        # Project onto plane
        K = self._compute_sectional(R, plane)
        
        return K

Lie Theory

Lie Groups

class LieGroup:
    """
    Lie group implementation.
    
    Theory:
        - [[lie_groups]]
        - [[lie_algebras]]
        - [[exponential_map]]
    Mathematics:
        - [[differential_geometry]]
        - [[group_theory]]
    """
    def __init__(self,
                 dimension: int,
                 multiplication: Callable):
        self.dim = dimension
        self.multiply = multiplication
        
    def lie_algebra_basis(self) -> List[np.ndarray]:
        """Get Lie algebra basis."""
        return self._compute_basis()
    
    def exponential(self,
                   X: np.ndarray) -> np.ndarray:
        """Compute Lie group exponential."""
        return self._compute_exponential(X)
    
    def adjoint(self,
               g: np.ndarray,
               X: np.ndarray) -> np.ndarray:
        """Compute adjoint action."""
        return self._compute_adjoint(g, X)

Principal Bundles

class PrincipalBundle:
    """
    Principal bundle implementation.
    
    Theory:
        - [[fiber_bundles]]
        - [[principal_connections]]
        - [[gauge_theory]]
    Mathematics:
        - [[differential_geometry]]
        - [[lie_theory]]
    """
    def __init__(self,
                 base: DifferentiableManifold,
                 structure_group: LieGroup):
        self.base = base
        self.group = structure_group
        
    def local_trivialization(self,
                           point: np.ndarray,
                           chart: str) -> Tuple[np.ndarray, np.ndarray]:
        """Get local trivialization."""
        return self._compute_trivialization(point, chart)
    
    def connection_form(self,
                       point: np.ndarray,
                       chart: str) -> np.ndarray:
        """Get connection 1-form."""
        return self._compute_connection_form(point, chart)

Applications to Active Inference

Belief Manifolds

class BeliefManifold:
    """
    Belief manifold implementation.
    
    Theory:
        - [[statistical_manifolds]]
        - [[information_geometry]]
        - [[belief_space]]
    Mathematics:
        - [[differential_geometry]]
        - [[probability_theory]]
    """
    def __init__(self,
                 dimension: int,
                 probability_model: ProbabilityModel):
        self.dim = dimension
        self.model = probability_model
        
    def fisher_metric(self,
                     belief: np.ndarray) -> np.ndarray:
        """Compute Fisher information metric."""
        return self._compute_fisher_metric(belief)
    
    def natural_gradient(self,
                        belief: np.ndarray,
                        gradient: np.ndarray) -> np.ndarray:
        """Compute natural gradient."""
        G = self.fisher_metric(belief)
        return np.linalg.solve(G, gradient)

Free Energy Geometry

class FreeEnergyGeometry:
    """
    Free energy geometric structure.
    
    Theory:
        - [[free_energy_principle]]
        - [[information_geometry]]
        - [[optimal_control]]
    Mathematics:
        - [[differential_geometry]]
        - [[symplectic_geometry]]
    """
    def __init__(self,
                 belief_manifold: BeliefManifold,
                 free_energy: Callable):
        self.manifold = belief_manifold
        self.F = free_energy
        
    def free_energy_metric(self,
                         belief: np.ndarray) -> np.ndarray:
        """Compute metric induced by free energy."""
        # Fisher metric
        G_fisher = self.manifold.fisher_metric(belief)
        
        # Free energy Hessian
        H = self._free_energy_hessian(belief)
        
        return G_fisher + H
    
    def hamiltonian_flow(self,
                        initial_belief: np.ndarray,
                        time: float) -> np.ndarray:
        """Compute Hamiltonian flow of free energy."""
        return self._solve_hamilton_equations(initial_belief, time)

Implementation Considerations

Numerical Methods

# @numerical_methods
numerical_implementations = {
    "geodesics": {
        "runge_kutta": "4th order RK method",
        "symplectic": "Symplectic integrators",
        "variational": "Variational integrators"
    },
    "curvature": {
        "finite_differences": "Numerical derivatives",
        "automatic_differentiation": "AD for tensors",
        "symbolic": "Symbolic computation"
    },
    "parallel_transport": {
        "schild": "Schild's ladder method",
        "pole": "Pole ladder method",
        "numerical": "Direct integration"
    }
}

Computational Efficiency

# @efficiency_considerations
efficiency_methods = {
    "metric_computation": {
        "caching": "Cache metric tensors",
        "approximation": "Low-rank approximations",
        "sparsity": "Exploit sparsity patterns"
    },
    "geodesic_computation": {
        "adaptive": "Adaptive step size",
        "local": "Local coordinate systems",
        "parallel": "Parallel transport methods"
    },
    "curvature_computation": {
        "lazy": "Lazy tensor evaluation",
        "symmetry": "Exploit symmetries",
        "distributed": "Parallel computation"
    }
}

Documentation Links

• ../../docs/research/research_documentation_index
• ../../docs/guides/implementation_guides_index
• ../../docs/api/api_documentation_index
• ../../docs/examples/usage_examples_index

References

• do_carmo - Riemannian Geometry
• lee - Introduction to Smooth Manifolds
• kobayashi_nomizu - Foundations of Differential Geometry
• marsden_ratiu - Introduction to Mechanics and Symmetry