Projective Geodesic Extensions

• Projective geodesic extensions are constructions that preserve unparameterized geodesic paths across various settings, including spray geometry, nonholonomic mechanics, and projective orbifolds.
• They enable reparameterization techniques that transform incomplete geodesics into complete ones by solving explicit ODEs along each curve.
• The framework integrates methods from affine connection theory, conformal modifications, and symmetry analysis to clarify relationships with invariant measures and Hamiltonization.

Projective geodesic extensions denote a family of constructions in which geodesic curves are preserved as unparameterized paths while the surrounding geometric data are modified. In spray geometry, two sprays are pointwise projectively related when $\bar G^i=G^i+P y^i$, so they have the same geodesics as point sets but generally with different parameterizations, and projective completeness asks whether the maximal parameter interval of each geodesic can be reparameterized onto $\mathbb R$ (Yang, 2023). In purely kinetic nonholonomic mechanics, a projective geodesic extension is a Riemannian metric $\hat g$ such that nonholonomic trajectories become pregeodesics (Belrhazi et al., 19 Sep 2025). In real projective geometry, the relevant extension phenomena appear at radial and totally geodesic ends, where projective geodesics are organized by a common vertex, a totally geodesic ideal boundary, or lens and horospherical models (Choi, 2013). This suggests that the expression names a common projective principle rather than a single standardized construction.

1. Common projective mechanism

The unifying mechanism is preservation of geodesics up to reparameterization. For sprays, the defining relation is

$\bar G^i = G^i + P y^i,$

with $P(x,y)$ the projective factor; the two sprays have the same geodesics as point sets, but not the same affine parameter (Yang, 2023). For Levi-Civita connections, a projective collineation is a vector field $X$ satisfying

$$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$

so that geodesics are mapped to geodesics, possibly with changed affine parameter (Paliathanasis, 2021). In non-symmetric affine geometry, the classical projective requirement is generalized to mappings that preserve a broader almost geodesic structure controlled by torsion and an affinor $F^i{}_j$ (Vesic, 2016).

A second recurrent theme is that “extension” may mean either reparameterization of already given curves or completion/compactification of the geometric setting in which those curves live. In real projective orbifolds, a radial end is foliated by properly embedded projective geodesics ending at a common point, while a totally geodesic end compactifies by adding a totally geodesic boundary orbifold (Choi, 2013). In that context, extension is tied to end geometry rather than only to parameter change.

Setting	Basic datum	Extension meaning
Spray geometry	$\bar G^i = G^i + P y^i$	Same geodesics, new parameter
Nonholonomic mechanics	$\hat g$ on $\mathbb R$0	Trajectories become pregeodesics
Real projective orbifolds	radial or totally geodesic end	Geodesics organized by vertex or ideal boundary

This range of usages is a frequent source of confusion. A common misconception is that projective geodesic extension always means adding points at infinity. The cited literature instead uses the phrase for reparameterized sprays, conformally modified nonholonomic metrics, and end compactifications in projective geometry.

2. Spray geometry and projective completeness

In spray geometry, a spray on $\mathbb R$1 is written in local coordinates $\mathbb R$2 as

$\mathbb R$3

with $\mathbb R$4 positively $\mathbb R$5-homogeneous in $\mathbb R$6. Its geodesics satisfy

$\mathbb R$7

If the same curve is reparameterized by $\mathbb R$8, then

$\mathbb R$9

which is the basic relation behind projective equivalence (Yang, 2023).

The classification of projectively flat sprays with weak Ricci constant or constant curvature is carried out at the level of geodesics. Theorem 1.1 states that along any geodesic the projective factor $\hat g$0 is one of three explicit forms, and in the complete case only the third type remains. Proposition 3.1 and Corollary 3.3 then give explicit admissible parameter changes $\hat g$1, including affine, logarithmic, arctangent, and logarithmic-ratio forms, with completeness restricting the allowable reparameterizations to the logarithmic cases. These results make the projective extension problem an explicit ODE problem along each geodesic rather than an abstract equivalence question.

A second major contribution is the path-space reconstruction method. A family of curves is an $\hat g$2-dimensional path space if through every tangent vector there is a path with that initial tangent, uniqueness holds locally for the same initial conditions, and the family is invariant under affine reparameterization $\hat g$3, $\hat g$4. Locally, every such path space can be written as

$\hat g$5

and Theorem 1.2 states that this induces a spray $\hat g$6 whose geodesics are exactly those curves with $\hat g$7 as geodesic parameter. If a new parameter $\hat g$8 satisfies $\hat g$9, then it gives a spray $\bar G^i = G^i + P y^i,$0 with $\bar G^i = G^i + P y^i,$1 as its geodesic parameter. In this setting, projective geodesic extension is literally a mechanism for passing from one spray to another through a common path space.

Projective completeness is addressed by Theorem 1.3. If every geodesic $\bar G^i = G^i + P y^i,$2 is defined on a maximal interval of one of the forms

$\bar G^i = G^i + P y^i,$3

with $\bar G^i = G^i + P y^i,$4 continuous on a conical region $\bar G^i = G^i + P y^i,$5, then the spray is projectively positively or negatively complete on $\bar G^i = G^i + P y^i,$6. The proof is constructive: one reparameterizes these intervals onto $\bar G^i = G^i + P y^i,$7 by logarithmic or finite-interval transformations such as

$\bar G^i = G^i + P y^i,$8

and then uses

$\bar G^i = G^i + P y^i,$9

to obtain a complete spray in the same projective class. The conceptual point is precise: incompleteness of the original affine parameter need not obstruct completeness of the projective class.

3. Nonholonomic mechanics and conformal modifications

For a purely kinetic nonholonomic system

$P(x,y)$0

a projective geodesic extension is a Riemannian metric $P(x,y)$1 such that the nonholonomic trajectories become pregeodesics of $P(x,y)$2 (Belrhazi et al., 19 Sep 2025). In projective-spray language, if $P(x,y)$3 and $P(x,y)$4 are sprays, they are projectively related if

$P(x,y)$5

where $P(x,y)$6 is the Liouville vector field and $P(x,y)$7 is $P(x,y)$8-homogeneous in the velocities; for quadratic sprays,

$P(x,y)$9

The problem is to find $X$0 and $X$1 so that the nonholonomic vector field equals the geodesic spray of $X$2 restricted to $X$3, up to the projective change $X$4.

The paper generalizes earlier $X$5-preserving modifications by allowing $X$6-conformal modifications: $X$7 with $X$8. The constrained Lagrangian is then preserved only up to conformal scaling,

$X$9

and the projective factor on $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$0 must satisfy

$$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$1

Lemma 3.1 gives necessary and sufficient conditions $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$2 and $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$3 for existence of such an extension: $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$4

$$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$5

where $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$6. Proposition 3.1 reduces the geometric existence problem to solving these conditions.

For Chaplygin systems, the paper clarifies the relation with $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$7-simplicity, invariant measures, and Hamiltonization. A Chaplygin system is $$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$8-simple if its gyroscopic tensor satisfies

$$\mathcal{L}_X \Gamma^i{}_{jk} = \delta^i{}_j\,\phi_{,k} + \delta^i{}_k\,\phi_{,j},$$9

or, in coordinates,

$F^i{}_j$0

Proposition 4.1 states that if the system is $F^i{}_j$1-simple, then

$F^i{}_j$2

gives a $F^i{}_j$3-orthogonal projective geodesic extension. In the $F^i{}_j$4-orthogonal case, $F^i{}_j$5 becomes redundant and the extension criterion reduces to $F^i{}_j$6.

The same paper shows that projective geodesic extension is strictly broader than $F^i{}_j$7-simplicity. Proposition 5.2 gives the classification:

1. geodesic extension with $F^i{}_j$8,
2. $F^i{}_j$9-simplicity,
3. projective geodesic extension via $\bar G^i = G^i + P y^i$0-conformal change,
4. invariant volume form.

The inclusion chain is

$\bar G^i = G^i + P y^i$1

but the reverse implications fail in general. The generalized nonholonomic particle, the two-wheeled carriage, and a $\bar G^i = G^i + P y^i$2-dimensional Chaplygin example are used to separate these notions. This directly addresses another common misconception: projective geodesic extensions are not equivalent to invariant measure or Hamiltonization, even though these structures are closely related.

4. Connection-based generalizations and projective invariants

In non-symmetric affine connection spaces $\bar G^i = G^i + P y^i$3, projective-geometric invariants can be extended from classical affine differential geometry to the setting of equitorsion second type almost geodesic mappings (Vesic, 2016). The ambient connection coefficients $\bar G^i = G^i + P y^i$4 are not assumed symmetric. The paper treats mappings $\bar G^i = G^i + P y^i$5 defined, in the equitorsion reciprocity case, by

$\bar G^i = G^i + P y^i$6

together with

$\bar G^i = G^i + P y^i$7

Here $\bar G^i = G^i + P y^i$8 is the affinor, and reciprocity is encoded by

$\bar G^i = G^i + P y^i$9

Equitorsion means

$\hat g$0

so the antisymmetric parts of the connections agree.

The invariant theory parallels the classical Thomas and Weyl projective tensors, but with torsion and mapping data included. Lemma 2.1 defines

$\hat g$1

and proves

$\hat g$2

This is the $\hat g$3-generalized Thomas projective parameter. The generalized Weyl projective tensor is built from the curvature of the associated symmetric spaces, together with trace and torsion-dependent correction terms, and Theorem 2.1 states that the tensor

$\hat g$4

defined in equation $\hat g$5 is invariant under the mapping.

The geometric meaning is explicit in the paper: in classical affine geometry, projective geometry is concerned with preserving geodesics up to reparametrization, whereas here second type almost geodesic mappings preserve a generalized geodesic structure determined by the affinor $\hat g$6 and torsion-dependent deformation terms. This extends projective curvature theory from torsion-free affine spaces to non-symmetric affine connection spaces without abandoning the projective emphasis on curve structure.

5. Convex projective manifolds, end theory, and geometric continuation

On a convex projective manifold $\hat g$7, where $\hat g$8 is a properly convex open set, the Hilbert metric is defined by

$\hat g$9

for aligned points $\mathbb R$00 (Blayac, 2020). In the non-strictly convex case, not every geodesic is straight, so the geodesic flow is defined using straight geodesics, namely intersections of $\mathbb R$01 with projective lines. This produces a projective geodesic dynamics whose natural recurrent part is the biproximal unit tangent bundle

$\mathbb R$02

If $\mathbb R$03 is strongly irreducible and $\mathbb R$04, then the geodesic flow is topologically mixing on $\mathbb R$05. In this setting, geodesic extension is organized by projective endpoints in $\mathbb R$06, especially by endpoints in the proximal limit set.

For strongly tame properly convex real projective orbifolds, the end theory is more rigid. A radial end is an end whose lifted neighborhood is foliated by properly embedded projective geodesics all ending at a common point, the p-end vertex. A totally geodesic end compactifies by adding a totally geodesic boundary orbifold. A lens-shaped domain $\mathbb R$07 has

$\mathbb R$08

with two smoothly strictly convex boundary hypersurfaces, and a lens-cone is $\mathbb R$09. A horospherical end is a radial end with a horoball neighborhood, and it is the projective analogue of a hyperbolic cusp (Choi, 2013).

The central classification theorem ties end geometry to end holonomy. For complete radial ends, one has

$\mathbb R$10

For properly convex radial or totally geodesic ends, the decisive hypothesis is the uniform middle-eigenvalue condition. Under strong tameness, strong irreducibility, admissible end groups, and the uniform middle-eigenvalue condition, a properly convex p-R-end is of generalized lens type, and a totally geodesic p-end is of lens type. Under weaker hypotheses, quasi-lens and quasi-joined behaviors appear (Choi, 2015).

End type	Local projective model	Classified behavior
Horospherical R-end	horoball neighborhood	complete affine $\mathbb R$11 cusp
Lens-shaped R-end	lens-cone $\mathbb R$12	uniform middle-eigenvalue $\mathbb R$13 generalized lens type
Totally geodesic end	ideal boundary $\mathbb R$14	uniform middle-eigenvalue $\mathbb R$15 lens type

These results show that extension near an end is not arbitrary. Geodesics may converge to a cusp point, be trapped between two strictly convex boundary components, or meet a totally geodesic ideal boundary. The NPCC case is more subtle: if an end is convex but not properly convex and not complete affine, then under the weakly uniform middle-eigenvalue condition it is of quasi-joined type, namely a quasi-join of a totally geodesic R-end and a cusp-type R-end (Choi, 2013). This is a genuine intermediate regime rather than a degeneration of the properly convex lens case.

6. Symmetry descriptions and projected real geodesics

Projective geodesic extension also appears as a symmetry problem. For the decomposable metric

$\mathbb R$16

Theorem 4 states that the Lie point symmetries for the geodesic equations of the $\mathbb R$17-dimensional Riemannian space $\mathbb R$18 form the projective algebra for the $\mathbb R$19-dimensional decomposable Riemannian space, and vice versa (Paliathanasis, 2021). The geodesic equations are

$\mathbb R$20

and the underlying reason is that the spatial part of the symmetry generator satisfies the same equations as a projective collineation of the Levi-Civita connection. In this formulation, an extension by one decomposable dimension geometrizes the Lie symmetry algebra of the geodesic ODEs.

A different construction starts from a complex manifold $\mathbb R$21 with Hermitian metric $\mathbb R$22 and projects to a real submanifold $\mathbb R$23. Theorem 1 derives the complex geodesic equation in holomorphic and anti-holomorphic coordinates, while Theorem 2 gives the projected geodesic on $\mathbb R$24: $\mathbb R$25 The projected motion contains the usual Christoffel-type term together with mixed $\mathbb R$26 terms, pure $\mathbb R$27 terms, link-tensor corrections, and antisymmetric field-strength contributions built from the imaginary part of the metric. If the metric is real symmetric and certain derivatives vanish, the extra terms disappear and the projective geodesic reduces to the usual real geodesic; in a special limit, the correction reduces to a Lorentz field structure (Sen, 2020).

A further continuation notion is the leaf extension of a complex submanifold $\mathbb R$28. The cited paper states explicitly that this construction “is not introduced as a geodesic extension in the Riemannian sense,” but behaves analogously as a maximal continuation principle in the projective category: the leaf extension is the maximal analytic continuation of $\mathbb R$29 through holomorphic immersions, and if the induced Kähler metric is extremal then the extension is complete (Li, 2023). This does not collapse the distinction between geodesic extension and analytic continuation, but it shows that projective continuation phenomena occur at several categorical levels.

Taken together, these strands show that projective geodesic extensions are best understood as a family of rigidifications of curve structure. They may act by reparameterizing sprays, conformally modifying constrained kinetic metrics, extracting projective invariants of non-symmetric connections, selecting dynamically relevant geodesics in convex projective manifolds, classifying the geometry of projective ends, or projecting complex geodesics onto real submanifolds. What remains constant across these settings is the projective viewpoint: the privileged object is the geodesic curve as a path, while metric, affine, dynamical, or boundary data are allowed to vary within a constrained projective class.