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Geodesics: Part 3: Core Theory

3. Core Theory

Core Theory develops the part of geodesics specified by the approved Chapter 25 table of contents. The treatment is geometry-first and AI-facing.

3.1 Christoffel symbols in coordinates

Christoffel symbols in coordinates belongs to the canonical scope of Geodesics. The goal is to make curved-space reasoning concrete enough for ML practice without turning the section into a pure topology course.

Working scope for this subsection: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview. The recurring pattern is localize, linearize, measure, move, and return to the manifold.

$$\frac{d^2 x^k}{dt^2}+\sum_{i,j}\Gamma^k_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt}=0.$$

Operational definition.

The Riemannian gradient is the tangent vector whose inner product with any direction equals the directional derivative.

Worked reading.

In coordinates with metric matrix $G$, the Riemannian gradient is $G^{-1} abla f$, not usually the raw Euclidean gradient.

Three examples of christoffel symbols in coordinates:

1. Natural gradient using Fisher information.
2. Projected gradient on the sphere.
3. Geometry-aware update for SPD covariance matrices.

Two non-examples clarify the boundary:

1. Raw parameter gradient treated as invariant under reparameterization.
2. A direction off the tangent space called a manifold gradient.

Proof or verification habit for christoffel symbols in coordinates:

Use the defining identity $g_p(\operatorname{grad} f,\mathbf{v})=df_p[\mathbf{v}]$ for all tangent directions.

global object      -> curved manifold or constraint set
local object       -> chart, tangent space, or coordinate patch
linear operation   -> derivative, gradient, velocity, Hessian approximation
geometric measure  -> metric, length, distance, curvature
algorithmic move   -> tangent step followed by geodesic or retraction

In AI systems, christoffel symbols in coordinates matters because learned representations and constrained parameter spaces are rarely globally flat. A local linear approximation may be useful, but it must be attached to the point where it is valid.

Natural gradient and second-order preconditioning are geometry choices, not only optimizer tricks.

Mini derivation lens.

1. Choose a point $p$ on the manifold $M$ and name the local representation used near $p$.
2. Move the question into a chart, tangent space, or embedded constraint where first-order calculus is available.
3. Compute the local object: derivative, tangent projection, metric-weighted gradient, path velocity, or retraction step.
4. Translate the result back into coordinate-free language so the answer is not tied to one chart by accident.
5. Check the invariant: the point remains on $M$, the direction remains in $T_pM$, or the distance/gradient uses the stated metric.

Implementation lens.

A practical ML implementation should store both the ambient array representation and the geometric contract attached to it. For example, a normalized embedding is not just a vector; it is a point on a sphere. An orthogonal weight matrix is not just a matrix; it is a point on a Stiefel-type constraint. A covariance matrix is not just a symmetric array; it must stay positive definite.

The clean computational pattern is: encode the state, compute an ambient derivative if needed, convert it into a tangent or metric-aware object, take a small local step, and then return to the manifold with a geodesic formula or retraction. This is the same pattern used in the companion notebooks, just scaled down to visible two- and three-dimensional examples.

The important warning is that coordinate code can pass shape checks while still violating geometry. Differential geometry adds checks that are semantic: tangentness, smooth compatibility, metric choice, path validity, and constraint preservation.

Practical checklist:

• State the manifold and whether it is abstract, embedded, or quotient-like.
• State the local coordinates or tangent representation being used.
• Separate ambient vectors from tangent vectors.
• Name the metric before computing distances, angles, or gradients.
• Use geodesics or retractions when moving on the manifold.
• For ML claims, identify whether geometry is data geometry, parameter geometry, or statistical geometry.

Local diagnostic: Ask which metric converts covectors into update vectors.

The companion notebook uses low-dimensional synthetic examples: circles, spheres, tangent projections, spherical interpolation, SPD matrices, and orthogonality constraints. These examples keep geometry visible while preserving the same update logic used in higher-dimensional ML systems.

A useful learning move is to compute everything first on a sphere. The sphere has visible curvature, simple tangent spaces, closed-form geodesics, and practical retractions. Once those are clear, Stiefel, Grassmann, SPD, and information-geometric examples become less mysterious.

For implementation, the main discipline is to avoid leaving the manifold silently. If a gradient step violates a constraint, either project the gradient into the tangent space before stepping or use a method whose update is intrinsic by design.

The final question for this subsection is whether a Euclidean formula is being used as an approximation, a coordinate expression, or a mistaken replacement for geometry. Differential geometry is the habit of telling those cases apart.

3.2 Geodesics on the sphere

Geodesics on the sphere belongs to the canonical scope of Geodesics. The goal is to make curved-space reasoning concrete enough for ML practice without turning the section into a pure topology course.

Working scope for this subsection: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview. The recurring pattern is localize, linearize, measure, move, and return to the manifold.

$$d_M(p,q)=\inf_{\gamma(0)=p,\gamma(1)=q} L(\gamma).$$

Operational definition.

A geodesic is a curve whose acceleration vanishes under the connection; locally, it is the curved-space analogue of a straight line.

Worked reading.

On a unit sphere, geodesics are great circles. The spherical interpolation formula stays on the sphere while linear interpolation cuts through the ambient ball.

Three examples of geodesics on the sphere:

1. Great-circle path between normalized embeddings.
2. Hyperbolic path through hierarchy embeddings.
3. Exponential-map step from a tangent vector.

Two non-examples clarify the boundary:

1. Ambient straight-line interpolation between two sphere points.
2. A shortest path across a discontinuous graph called a smooth geodesic.

Proof or verification habit for geodesics on the sphere:

Check the geodesic equation or use known symmetry of the manifold to characterize the path.

global object      -> curved manifold or constraint set
local object       -> chart, tangent space, or coordinate patch
linear operation   -> derivative, gradient, velocity, Hessian approximation
geometric measure  -> metric, length, distance, curvature
algorithmic move   -> tangent step followed by geodesic or retraction

In AI systems, geodesics on the sphere matters because learned representations and constrained parameter spaces are rarely globally flat. A local linear approximation may be useful, but it must be attached to the point where it is valid.

Geodesics make latent interpolation, representation distances, and motion planning respect the actual geometry.

Mini derivation lens.

1. Choose a point $p$ on the manifold $M$ and name the local representation used near $p$.
2. Move the question into a chart, tangent space, or embedded constraint where first-order calculus is available.
3. Compute the local object: derivative, tangent projection, metric-weighted gradient, path velocity, or retraction step.
4. Translate the result back into coordinate-free language so the answer is not tied to one chart by accident.
5. Check the invariant: the point remains on $M$, the direction remains in $T_pM$, or the distance/gradient uses the stated metric.

Implementation lens.

A practical ML implementation should store both the ambient array representation and the geometric contract attached to it. For example, a normalized embedding is not just a vector; it is a point on a sphere. An orthogonal weight matrix is not just a matrix; it is a point on a Stiefel-type constraint. A covariance matrix is not just a symmetric array; it must stay positive definite.

The clean computational pattern is: encode the state, compute an ambient derivative if needed, convert it into a tangent or metric-aware object, take a small local step, and then return to the manifold with a geodesic formula or retraction. This is the same pattern used in the companion notebooks, just scaled down to visible two- and three-dimensional examples.

The important warning is that coordinate code can pass shape checks while still violating geometry. Differential geometry adds checks that are semantic: tangentness, smooth compatibility, metric choice, path validity, and constraint preservation.

Practical checklist:

• State the manifold and whether it is abstract, embedded, or quotient-like.
• State the local coordinates or tangent representation being used.
• Separate ambient vectors from tangent vectors.
• Name the metric before computing distances, angles, or gradients.
• Use geodesics or retractions when moving on the manifold.
• For ML claims, identify whether geometry is data geometry, parameter geometry, or statistical geometry.

Local diagnostic: Verify the path stays on the manifold and has the right initial velocity.

The companion notebook uses low-dimensional synthetic examples: circles, spheres, tangent projections, spherical interpolation, SPD matrices, and orthogonality constraints. These examples keep geometry visible while preserving the same update logic used in higher-dimensional ML systems.

A useful learning move is to compute everything first on a sphere. The sphere has visible curvature, simple tangent spaces, closed-form geodesics, and practical retractions. Once those are clear, Stiefel, Grassmann, SPD, and information-geometric examples become less mysterious.

For implementation, the main discipline is to avoid leaving the manifold silently. If a gradient step violates a constraint, either project the gradient into the tangent space before stepping or use a method whose update is intrinsic by design.

The final question for this subsection is whether a Euclidean formula is being used as an approximation, a coordinate expression, or a mistaken replacement for geometry. Differential geometry is the habit of telling those cases apart.

3.3 Geodesic distance and metric completeness

Geodesic distance and metric completeness belongs to the canonical scope of Geodesics. The goal is to make curved-space reasoning concrete enough for ML practice without turning the section into a pure topology course.

Working scope for this subsection: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview. The recurring pattern is localize, linearize, measure, move, and return to the manifold.

$$\dot{\gamma}(t)\in T_{\gamma(t)}M.$$

Operational definition.

A Riemannian metric assigns an inner product to every tangent space smoothly.

Worked reading.

If a coordinate metric is $G(\mathbf{x})$, then length of a velocity $\dot{\mathbf{x}}$ is $\sqrt{\dot{\mathbf{x}}^\top G(\mathbf{x})\dot{\mathbf{x}}}$.

Three examples of geodesic distance and metric completeness:

1. Euclidean metric on a sphere inherited from ambient space.
2. Fisher metric on statistical models.
3. Affine-invariant metric on SPD matrices.

Two non-examples clarify the boundary:

1. A distance formula with no tangent-space inner product.
2. A fixed Euclidean metric used after nonlinear reparameterization without checking geometry.

Proof or verification habit for geodesic distance and metric completeness:

Check symmetry, bilinearity, positive definiteness, and smooth variation with the base point.

global object      -> curved manifold or constraint set
local object       -> chart, tangent space, or coordinate patch
linear operation   -> derivative, gradient, velocity, Hessian approximation
geometric measure  -> metric, length, distance, curvature
algorithmic move   -> tangent step followed by geodesic or retraction

In AI systems, geodesic distance and metric completeness matters because learned representations and constrained parameter spaces are rarely globally flat. A local linear approximation may be useful, but it must be attached to the point where it is valid.

The metric determines what steepest descent, distance, and regularization mean for a representation or parameter space.

Mini derivation lens.

1. Choose a point $p$ on the manifold $M$ and name the local representation used near $p$.
2. Move the question into a chart, tangent space, or embedded constraint where first-order calculus is available.
3. Compute the local object: derivative, tangent projection, metric-weighted gradient, path velocity, or retraction step.
4. Translate the result back into coordinate-free language so the answer is not tied to one chart by accident.
5. Check the invariant: the point remains on $M$, the direction remains in $T_pM$, or the distance/gradient uses the stated metric.

Implementation lens.

A practical ML implementation should store both the ambient array representation and the geometric contract attached to it. For example, a normalized embedding is not just a vector; it is a point on a sphere. An orthogonal weight matrix is not just a matrix; it is a point on a Stiefel-type constraint. A covariance matrix is not just a symmetric array; it must stay positive definite.

The clean computational pattern is: encode the state, compute an ambient derivative if needed, convert it into a tangent or metric-aware object, take a small local step, and then return to the manifold with a geodesic formula or retraction. This is the same pattern used in the companion notebooks, just scaled down to visible two- and three-dimensional examples.

The important warning is that coordinate code can pass shape checks while still violating geometry. Differential geometry adds checks that are semantic: tangentness, smooth compatibility, metric choice, path validity, and constraint preservation.

Practical checklist:

• State the manifold and whether it is abstract, embedded, or quotient-like.
• State the local coordinates or tangent representation being used.
• Separate ambient vectors from tangent vectors.
• Name the metric before computing distances, angles, or gradients.
• Use geodesics or retractions when moving on the manifold.
• For ML claims, identify whether geometry is data geometry, parameter geometry, or statistical geometry.

Local diagnostic: State the metric before computing lengths or gradients.

The companion notebook uses low-dimensional synthetic examples: circles, spheres, tangent projections, spherical interpolation, SPD matrices, and orthogonality constraints. These examples keep geometry visible while preserving the same update logic used in higher-dimensional ML systems.

A useful learning move is to compute everything first on a sphere. The sphere has visible curvature, simple tangent spaces, closed-form geodesics, and practical retractions. Once those are clear, Stiefel, Grassmann, SPD, and information-geometric examples become less mysterious.

For implementation, the main discipline is to avoid leaving the manifold silently. If a gradient step violates a constraint, either project the gradient into the tangent space before stepping or use a method whose update is intrinsic by design.

The final question for this subsection is whether a Euclidean formula is being used as an approximation, a coordinate expression, or a mistaken replacement for geometry. Differential geometry is the habit of telling those cases apart.

3.4 Parallel transport preview

Parallel transport preview belongs to the canonical scope of Geodesics. The goal is to make curved-space reasoning concrete enough for ML practice without turning the section into a pure topology course.

Working scope for this subsection: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview. The recurring pattern is localize, linearize, measure, move, and return to the manifold.

$$\nabla_{\dot{\gamma}}\dot{\gamma}=0.$$

Operational definition.

Energy, length, transport, and geodesic convexity are tools for comparing paths and moving tangent information across points.

Worked reading.

A tangent vector at one point cannot be subtracted from a tangent vector at another point without a transport rule.

Three examples of parallel transport preview:

1. Transporting momentum in Riemannian optimization.
2. Geodesically convex SPD objectives.
3. Shortest-path loss for robotics trajectories.

Two non-examples clarify the boundary:

1. Subtracting tangent vectors at different points as if all tangent spaces were identical.
2. Euclidean convexity assumed on a curved domain.

Proof or verification habit for parallel transport preview:

The proof habit is to phrase inequalities along geodesics, not ambient line segments.

global object      -> curved manifold or constraint set
local object       -> chart, tangent space, or coordinate patch
linear operation   -> derivative, gradient, velocity, Hessian approximation
geometric measure  -> metric, length, distance, curvature
algorithmic move   -> tangent step followed by geodesic or retraction

In AI systems, parallel transport preview matters because learned representations and constrained parameter spaces are rarely globally flat. A local linear approximation may be useful, but it must be attached to the point where it is valid.

This is where geometry changes training dynamics: momentum, convexity, and interpolation all need curved-space versions.

Mini derivation lens.

1. Choose a point $p$ on the manifold $M$ and name the local representation used near $p$.
2. Move the question into a chart, tangent space, or embedded constraint where first-order calculus is available.
3. Compute the local object: derivative, tangent projection, metric-weighted gradient, path velocity, or retraction step.
4. Translate the result back into coordinate-free language so the answer is not tied to one chart by accident.
5. Check the invariant: the point remains on $M$, the direction remains in $T_pM$, or the distance/gradient uses the stated metric.

Implementation lens.

A practical ML implementation should store both the ambient array representation and the geometric contract attached to it. For example, a normalized embedding is not just a vector; it is a point on a sphere. An orthogonal weight matrix is not just a matrix; it is a point on a Stiefel-type constraint. A covariance matrix is not just a symmetric array; it must stay positive definite.

The clean computational pattern is: encode the state, compute an ambient derivative if needed, convert it into a tangent or metric-aware object, take a small local step, and then return to the manifold with a geodesic formula or retraction. This is the same pattern used in the companion notebooks, just scaled down to visible two- and three-dimensional examples.

The important warning is that coordinate code can pass shape checks while still violating geometry. Differential geometry adds checks that are semantic: tangentness, smooth compatibility, metric choice, path validity, and constraint preservation.

Practical checklist:

• State the manifold and whether it is abstract, embedded, or quotient-like.
• State the local coordinates or tangent representation being used.
• Separate ambient vectors from tangent vectors.
• Name the metric before computing distances, angles, or gradients.
• Use geodesics or retractions when moving on the manifold.
• For ML claims, identify whether geometry is data geometry, parameter geometry, or statistical geometry.

Local diagnostic: Ask whether the comparison happens inside one tangent space or across different tangent spaces.

The companion notebook uses low-dimensional synthetic examples: circles, spheres, tangent projections, spherical interpolation, SPD matrices, and orthogonality constraints. These examples keep geometry visible while preserving the same update logic used in higher-dimensional ML systems.

A useful learning move is to compute everything first on a sphere. The sphere has visible curvature, simple tangent spaces, closed-form geodesics, and practical retractions. Once those are clear, Stiefel, Grassmann, SPD, and information-geometric examples become less mysterious.

For implementation, the main discipline is to avoid leaving the manifold silently. If a gradient step violates a constraint, either project the gradient into the tangent space before stepping or use a method whose update is intrinsic by design.

The final question for this subsection is whether a Euclidean formula is being used as an approximation, a coordinate expression, or a mistaken replacement for geometry. Differential geometry is the habit of telling those cases apart.

3.5 Geodesic convexity preview

Geodesic convexity preview belongs to the canonical scope of Geodesics. The goal is to make curved-space reasoning concrete enough for ML practice without turning the section into a pure topology course.

Working scope for this subsection: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview. The recurring pattern is localize, linearize, measure, move, and return to the manifold.

$$\frac{d^2 x^k}{dt^2}+\sum_{i,j}\Gamma^k_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt}=0.$$

Operational definition.

A geodesic is a curve whose acceleration vanishes under the connection; locally, it is the curved-space analogue of a straight line.

Worked reading.

On a unit sphere, geodesics are great circles. The spherical interpolation formula stays on the sphere while linear interpolation cuts through the ambient ball.

Three examples of geodesic convexity preview:

1. Great-circle path between normalized embeddings.
2. Hyperbolic path through hierarchy embeddings.
3. Exponential-map step from a tangent vector.

Two non-examples clarify the boundary:

1. Ambient straight-line interpolation between two sphere points.
2. A shortest path across a discontinuous graph called a smooth geodesic.

Proof or verification habit for geodesic convexity preview:

Check the geodesic equation or use known symmetry of the manifold to characterize the path.

global object      -> curved manifold or constraint set
local object       -> chart, tangent space, or coordinate patch
linear operation   -> derivative, gradient, velocity, Hessian approximation
geometric measure  -> metric, length, distance, curvature
algorithmic move   -> tangent step followed by geodesic or retraction

In AI systems, geodesic convexity preview matters because learned representations and constrained parameter spaces are rarely globally flat. A local linear approximation may be useful, but it must be attached to the point where it is valid.

Geodesics make latent interpolation, representation distances, and motion planning respect the actual geometry.

Mini derivation lens.

1. Choose a point $p$ on the manifold $M$ and name the local representation used near $p$.
2. Move the question into a chart, tangent space, or embedded constraint where first-order calculus is available.
3. Compute the local object: derivative, tangent projection, metric-weighted gradient, path velocity, or retraction step.
4. Translate the result back into coordinate-free language so the answer is not tied to one chart by accident.
5. Check the invariant: the point remains on $M$, the direction remains in $T_pM$, or the distance/gradient uses the stated metric.

Implementation lens.

A practical ML implementation should store both the ambient array representation and the geometric contract attached to it. For example, a normalized embedding is not just a vector; it is a point on a sphere. An orthogonal weight matrix is not just a matrix; it is a point on a Stiefel-type constraint. A covariance matrix is not just a symmetric array; it must stay positive definite.

The clean computational pattern is: encode the state, compute an ambient derivative if needed, convert it into a tangent or metric-aware object, take a small local step, and then return to the manifold with a geodesic formula or retraction. This is the same pattern used in the companion notebooks, just scaled down to visible two- and three-dimensional examples.

The important warning is that coordinate code can pass shape checks while still violating geometry. Differential geometry adds checks that are semantic: tangentness, smooth compatibility, metric choice, path validity, and constraint preservation.

Practical checklist:

• State the manifold and whether it is abstract, embedded, or quotient-like.
• State the local coordinates or tangent representation being used.
• Separate ambient vectors from tangent vectors.
• Name the metric before computing distances, angles, or gradients.
• Use geodesics or retractions when moving on the manifold.
• For ML claims, identify whether geometry is data geometry, parameter geometry, or statistical geometry.

Local diagnostic: Verify the path stays on the manifold and has the right initial velocity.

The companion notebook uses low-dimensional synthetic examples: circles, spheres, tangent projections, spherical interpolation, SPD matrices, and orthogonality constraints. These examples keep geometry visible while preserving the same update logic used in higher-dimensional ML systems.

A useful learning move is to compute everything first on a sphere. The sphere has visible curvature, simple tangent spaces, closed-form geodesics, and practical retractions. Once those are clear, Stiefel, Grassmann, SPD, and information-geometric examples become less mysterious.

For implementation, the main discipline is to avoid leaving the manifold silently. If a gradient step violates a constraint, either project the gradient into the tangent space before stepping or use a method whose update is intrinsic by design.

The final question for this subsection is whether a Euclidean formula is being used as an approximation, a coordinate expression, or a mistaken replacement for geometry. Differential geometry is the habit of telling those cases apart.