# 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|Research Documentation]]
- [[information_geometry]]
- type: uses
links:
- [[tensor_calculus]]
- [[lie_theory]]
- type: documented_by
links:
- [[../../docs/guides/implementation_guides_index|Implementation Guides]]
- [[../../docs/api/api_documentation_index|API Documentation]]
---
## 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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
# @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
```python
# @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|Research Documentation]]
- [[../../docs/guides/implementation_guides_index|Implementation Guides]]
- [[../../docs/api/api_documentation_index|API Documentation]]
- [[../../docs/examples/usage_examples_index|Usage Examples]]
## References
- [[do_carmo]] - Riemannian Geometry
- [[lee]] - Introduction to Smooth Manifolds
- [[kobayashi_nomizu]] - Foundations of Differential Geometry
- [[marsden_ratiu]] - Introduction to Mechanics and Symmetry
---
title: Differential Geometry
type: concept
status: stable
created: 2024-02-12
tags:
- mathematics
- geometry
- topology
semantic_relations:
- type: foundation
links:
- [[manifold_theory]]
- [[topology]]
- type: relates
links:
- [[information_geometry]]
- [[lie_groups]]
- [[tensor_analysis]]
---
# Differential Geometry
## Core Concepts
### Manifolds
1. **Smooth Manifolds**
```math
M = \{(U_α,φ_α) | α ∈ A\}
```
where:
- U_α are open sets
- φ_α are coordinate charts
- A is atlas index set
2. **Tangent Space**
```math
T_pM = span\{\frac{∂}{∂x^i}|_p\}
```
where:
- p is point on manifold
- x^i are local coordinates
### Differential Forms
1. **Exterior Derivative**
```math
dω = \sum_{i_1<...<i_k} \frac{∂ω_{i_1...i_k}}{∂x^j}dx^j ∧ dx^{i_1} ∧ ... ∧ dx^{i_k}
```
where:
- ω is k-form
- ∧ is wedge product
2. **Integration**
```math
\int_M ω = \sum_α \int_{φ_α(U_α)} (φ_α^{-1})^*ω
```
where:
- ω is n-form
- φ_α are charts
### Riemannian Geometry
1. **Metric Tensor**
```math
ds² = g_{ij}dx^idx^j
```
where:
- g_{ij} is metric components
- dx^i are coordinate differentials
2. **Christoffel Symbols**
```math
Γ^k_{ij} = \frac{1}{2}g^{kl}(∂_ig_{jl} + ∂_jg_{il} - ∂_lg_{ij})
```
where:
- g^{kl} is inverse metric
- ∂_i is partial derivative
## Advanced Concepts
### Connection Theory
1. **Covariant Derivative**
```math
∇_X Y = (X^i∂_iY^k + Γ^k_{ij}X^iY^j)\frac{∂}{∂x^k}
```
where:
- X,Y are vector fields
- Γ^k_{ij} are connection coefficients
2. **Parallel Transport**
```math
\frac{D}{dt}V^i + Γ^i_{jk}\frac{dx^j}{dt}V^k = 0
```
where:
- V^i are vector components
- D/dt is covariant derivative
### Curvature
1. **Riemann Tensor**
```math
R^i_{jkl} = ∂_kΓ^i_{jl} - ∂_lΓ^i_{jk} + Γ^i_{mk}Γ^m_{jl} - Γ^i_{ml}Γ^m_{jk}
```
where:
- Γ^i_{jk} are Christoffel symbols
2. **Ricci Tensor**
```math
R_{ij} = R^k_{ikj}
```
where:
- R^k_{ikj} is Riemann tensor
### Lie Groups
1. **Lie Algebra**
```math
[X,Y] = XY - YX
```
where:
- X,Y are vector fields
- [,] is Lie bracket
2. **Exponential Map**
```math
exp: g → G, exp(X) = \sum_{n=0}^∞ \frac{X^n}{n!}
```
where:
- g is Lie algebra
- G is Lie group
## Applications
### Information Geometry
1. **Fisher Metric**
```math
g_{ij}(θ) = E[-\frac{∂²}{∂θ^i∂θ^j}log p(x|θ)]
```
where:
- θ are statistical parameters
- p(x|θ) is probability model
2. **α-Connections**
```math
Γ^{(α)}_{ijk} = E[\frac{∂}{\∂θ^i}log p(x|θ)\frac{∂}{\∂θ^j}log p(x|θ)\frac{∂}{\∂θ^k}log p(x|θ)]
```
where:
- α is connection parameter
### General Relativity
1. **Einstein Field Equations**
```math
R_{μν} - \frac{1}{2}Rg_{μν} + Λg_{μν} = \frac{8πG}{c^4}T_{μν}
```
where:
- R_{μν} is Ricci tensor
- T_{μν} is stress-energy tensor
2. **Geodesic Equation**
```math
\frac{d²x^μ}{dτ²} + Γ^μ_{αβ}\frac{dx^α}{dτ}\frac{dx^β}{dτ} = 0
```
where:
- τ is proper time
- Γ^μ_{αβ} are Christoffel symbols
### Machine Learning
1. **Natural Gradient**
```math
θ_{t+1} = θ_t - ηg^{ij}(θ_t)∂_jL(θ_t)
```
where:
- g^{ij} is inverse Fisher metric
- L is loss function
2. **Manifold Learning**
```math
min_f \int_M ||∇f||²_g dV_g
```
where:
- f is embedding
- g is metric
- dV_g is volume form
## Implementation
### Differential Geometry Tools
```python
class DifferentialGeometry:
def __init__(self,
manifold: Manifold,
metric: Metric):
"""Initialize differential geometry tools.
Args:
manifold: Manifold structure
metric: Riemannian metric
"""
self.manifold = manifold
self.metric = metric
def christoffel_symbols(self,
point: np.ndarray) -> np.ndarray:
"""Compute Christoffel symbols.
Args:
point: Point on manifold
Returns:
gamma: Christoffel symbols
"""
# Get metric and derivatives
g = self.metric.evaluate(point)
dg = self.metric.derivative(point)
# Compute inverse metric
g_inv = np.linalg.inv(g)
# Compute Christoffel symbols
gamma = np.zeros((g.shape[0], g.shape[0], g.shape[0]))
for i in range(g.shape[0]):
for j in range(g.shape[0]):
for k in range(g.shape[0]):
gamma[i,j,k] = 0.5 * np.sum(
g_inv[i,:] * (
dg[j,:,k] + dg[k,:,j] - dg[:,j,k]
)
)
return gamma
```
### Geodesic Integration
```python
class GeodesicIntegrator:
def __init__(self,
geometry: DifferentialGeometry,
step_size: float = 0.01):
"""Initialize geodesic integrator.
Args:
geometry: Differential geometry tools
step_size: Integration step size
"""
self.geometry = geometry
self.step_size = step_size
def integrate(self,
initial_point: np.ndarray,
initial_velocity: np.ndarray,
n_steps: int) -> Tuple[np.ndarray, np.ndarray]:
"""Integrate geodesic equation.
Args:
initial_point: Starting point
initial_velocity: Initial velocity
n_steps: Number of integration steps
Returns:
points: Geodesic points
velocities: Geodesic velocities
"""
points = [initial_point]
velocities = [initial_velocity]
for _ in range(n_steps):
# Get current state
point = points[-1]
velocity = velocities[-1]
# Compute Christoffel symbols
gamma = self.geometry.christoffel_symbols(point)
# Update velocity
acceleration = -np.sum(
gamma * velocity[None,:] * velocity[:,None],
axis=(0,1)
)
new_velocity = velocity + self.step_size * acceleration
# Update position
new_point = point + self.step_size * velocity
# Store results
points.append(new_point)
velocities.append(new_velocity)
return np.array(points), np.array(velocities)
```
## Advanced Topics
### Symplectic Geometry
1. **Symplectic Form**
```math
ω = dx^i ∧ dp_i
```
where:
- x^i are position coordinates
- p_i are momentum coordinates
2. **Hamiltonian Flow**
```math
\dot{x}^i = \frac{∂H}{∂p_i}, \dot{p}_i = -\frac{∂H}{∂x^i}
```
where:
- H is Hamiltonian
- x^i,p_i are canonical coordinates
### Complex Geometry
1. **Kähler Manifolds**
```math
ω = ig_{αβ̄}dz^α ∧ dz̄^β
```
where:
- g_{αβ̄} is Hermitian metric
- z^α are complex coordinates
2. **Holomorphic Forms**
```math
∂ω = 0
```
where:
- ∂ is Dolbeault operator
- ω is differential form
## Future Directions
### Emerging Areas
1. **Geometric Deep Learning**
- Group equivariant networks
- Manifold optimization
- Topological data analysis
2. **Quantum Geometry**
- Non-commutative geometry
- Quantum gravity
- String theory
### Open Problems
1. **Theoretical Challenges**
- Geometric flows
- Singular geometries
- Mirror symmetry
2. **Practical Challenges**
- Numerical methods
- High-dimensional manifolds
- Discrete geometry
## Related Topics
1. [[topology|Topology]]
2. [[lie_theory|Lie Theory]]
3. [[algebraic_geometry|Algebraic Geometry]]
4. [[geometric_analysis|Geometric Analysis]]