Compute Curvature

Compute geometric curvature quantities from a Riemannian metric.

When to Use

•User wants to analyze the curvature of a manifold
•User needs Christoffel symbols for geodesic equations
•User wants to compute Riemann, Ricci, or scalar curvature
•User wants to check if a space is flat
•User mentions "connection", "curvature", "Christoffel"

Key Imports

python

import jax
import jax.numpy as jnp
from local_coordinates.jet import function_to_jet
from local_coordinates.basis import get_standard_basis
from local_coordinates.metric import RiemannianMetric
from local_coordinates.connection import Connection, get_levi_civita_connection
from local_coordinates.riemann import (
    RiemannCurvatureTensor,
    get_riemann_curvature_tensor,
    RicciTensor,
    get_ricci_tensor
)

Prerequisites

python

jax.config.update("jax_enable_x64", True)

Instructions

Step 1: Create a Metric

First, create a RiemannianMetric object (see create-riemannian-metric skill):

python

def metric_components(x):
    return jnp.array([
        [1 + 0.1*x[0]**2, 0.0],
        [0.0, 1 + 0.1*x[1]**2]
    ])

p = jnp.array([1.0, 1.0])
basis = get_standard_basis(p)
metric_jet = function_to_jet(metric_components, p)
metric = RiemannianMetric(basis=basis, components=metric_jet)

Step 2: Compute the Levi-Civita Connection

The Levi-Civita connection is the unique torsion-free, metric-compatible connection:

python

connection = get_levi_civita_connection(metric)

Step 3: Access Christoffel Symbols

The Christoffel symbols are indexed as Gamma^k_ij = christoffel_symbols[i, j, k]:

python

Gamma = connection.christoffel_symbols.value
print(f"Christoffel symbols shape: {Gamma.shape}")  # (D, D, D)
print(f"Gamma^0_00 = {Gamma[0, 0, 0]}")

Step 4: Compute Riemann Curvature Tensor

The (3,1) Riemann curvature tensor R^m_ijk:

python

riemann = get_riemann_curvature_tensor(connection)
R = riemann.components.value
print(f"Riemann tensor shape: {R.shape}")  # (D, D, D, D)

Index convention: R[i, j, k, m] = R^m_ijk

Step 5: Compute Ricci Tensor

The Ricci tensor is the contraction R_ab = R^i_aib:

python

ricci = get_ricci_tensor(connection, R=riemann)
Ric = ricci.components.value
print(f"Ricci tensor:\n{Ric}")

Step 6: Compute Scalar Curvature

The scalar curvature is R = g^ij R_ij:

python

g_inv = jnp.linalg.inv(metric.components.value)
scalar_curvature = jnp.einsum("ij,ij->", g_inv, ricci.components.value)
print(f"Scalar curvature: {scalar_curvature}")

Complete Example

python

import jax
import jax.numpy as jnp
from local_coordinates.jet import function_to_jet
from local_coordinates.basis import get_standard_basis
from local_coordinates.metric import RiemannianMetric
from local_coordinates.connection import get_levi_civita_connection
from local_coordinates.riemann import get_riemann_curvature_tensor, get_ricci_tensor

jax.config.update("jax_enable_x64", True)

# Define metric
def metric_components(x):
    return jnp.array([
        [1 + 0.1*x[0]**2, 0.0],
        [0.0, 1 + 0.1*x[1]**2]
    ])

p = jnp.array([1.0, 1.0])
basis = get_standard_basis(p)
metric_jet = function_to_jet(metric_components, p)
metric = RiemannianMetric(basis=basis, components=metric_jet)

# Compute all curvature quantities
connection = get_levi_civita_connection(metric)
riemann = get_riemann_curvature_tensor(connection)
ricci = get_ricci_tensor(connection, R=riemann)

# Scalar curvature
g_inv = jnp.linalg.inv(metric.components.value)
scalar_curvature = jnp.einsum("ij,ij->", g_inv, ricci.components.value)

print(f"Metric at p:\n{metric.components.value}")
print(f"Christoffel symbols:\n{connection.christoffel_symbols.value}")
print(f"Ricci tensor:\n{ricci.components.value}")
print(f"Scalar curvature: {scalar_curvature}")

Checking Flatness

A space is flat if the Riemann tensor vanishes:

python

is_flat = jnp.allclose(riemann.components.value, 0.0, atol=1e-10)
print(f"Space is flat: {is_flat}")

Using Covariant Derivatives

The connection provides covariant derivatives:

python

from local_coordinates.tangent import TangentVector
from local_coordinates.jet import Jet

# Create tangent vectors X and Y
v_jet = Jet(
    value=jnp.array([1.0, 0.0]),
    gradient=jnp.zeros((2, 2)),
    hessian=jnp.zeros((2, 2, 2))
)
X = TangentVector(p=p, components=v_jet, basis=basis)
Y = TangentVector(p=p, components=v_jet, basis=basis)

# Compute covariant derivative nabla_X Y
nabla_X_Y = connection.covariant_derivative(X, Y)
print(f"Covariant derivative components: {nabla_X_Y.components.value}")

Riemann Tensor Symmetries

The Riemann tensor satisfies:

•Skew symmetry: R_ijkl = -R_jikl
•Interchange symmetry: R_ijkl = R_klij
•First Bianchi identity: R_ijkl + R_jkil + R_kijl = 0

Important Notes

•The Riemann tensor is a (3,1) tensor with index convention R[i, j, k, m] = R^m_ijk
•The Christoffel symbols are NOT tensors and transform specially under coordinate changes
•For flat space (Euclidean), all curvature quantities vanish
•Ricci tensor is symmetric: R_ij = R_ji