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Geodesics: Part 2: Formal Definitions

2. Formal Definitions

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

2.1 Curves $\gamma(t)$ on manifolds

Curves $\gamma(t)$ on manifolds 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.

Curves $\gamma(t)$ on manifolds belongs to the canonical scope of Geodesics: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview.

Worked reading.

Start from a concrete embedded example, compute the local tangent or metric object, then translate back to intrinsic notation.

Three examples of curves $\gamma(t)$ on manifolds:

1. Sphere geometry.
2. Embedding-space local coordinates.
3. Matrix-manifold parameter constraints.

Two non-examples clarify the boundary:

1. A flat Euclidean approximation used globally.
2. A geometric claim made without metric or tangent space.

Proof or verification habit for curves $\gamma(t)$ on manifolds:

The proof habit is to compute locally and verify coordinate-independent meaning.

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, curves $\gamma(t)$ on manifolds 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 AI relevance is that model spaces are often curved even when implemented as arrays.

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: Name the manifold, tangent space, metric, and map being used.

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.

2.2 Velocity field $\dot{\gamma}(t)\in T_{\gamma(t)}M$

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

Velocity field $\dot{\gamma}(t)\in T_{\gamma(t)}M$ belongs to the canonical scope of Geodesics: curves, velocities, geodesic equation, exponential and logarithm maps, Christoffel symbols, sphere geodesics, parallel transport, and geodesic convexity preview.

Worked reading.

Start from a concrete embedded example, compute the local tangent or metric object, then translate back to intrinsic notation.

Three examples of velocity field $\dot{\gamma}(t)\in t_{\gamma(t)}m$:

1. Sphere geometry.
2. Embedding-space local coordinates.
3. Matrix-manifold parameter constraints.

Two non-examples clarify the boundary:

1. A flat Euclidean approximation used globally.
2. A geometric claim made without metric or tangent space.

Proof or verification habit for velocity field $\dot{\gamma}(t)\in t_{\gamma(t)}m$:

The proof habit is to compute locally and verify coordinate-independent meaning.

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, velocity field $\dot{\gamma}(t)\in t_{\gamma(t)}m$ 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 AI relevance is that model spaces are often curved even when implemented as arrays.

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: Name the manifold, tangent space, metric, and map being used.

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.

2.3 Geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$

Geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$ 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 geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$:

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 equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$:

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

2.4 Exponential map $\operatorname{Exp}_p(\mathbf{v})$

Exponential map $\operatorname{Exp}_p(\mathbf{v})$ 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 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 exponential map $\operatorname{exp}_p(\mathbf{v})$:

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 exponential map $\operatorname{exp}_p(\mathbf{v})$:

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, exponential map $\operatorname{exp}_p(\mathbf{v})$ 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.

2.5 Logarithm map and injectivity-radius preview

Logarithm map and injectivity-radius 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.

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 logarithm map and injectivity-radius 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 logarithm map and injectivity-radius 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, logarithm map and injectivity-radius 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.