Wind Finslerian Structures in Differential Geometry

• Wind Finslerian structures are generalized Finsler metrics characterized by strongly convex tangent indicatrices that need not contain the origin, enabling advanced geometric modeling.
• They provide a unified framework for addressing the Zermelo navigation problem under arbitrary winds and analyzing causal properties in Lorentzian spacetimes with Killing fields.
• This framework links Randers and Kropina metrics, offering practical insights into geodesic completeness, causality, and optimal navigational solutions in complex geometries.

A wind Finslerian structure generalizes the concept of a Finsler metric by permitting strongly convex, compact indicatrices in each tangent space that need not contain the origin. This provides a unifying geometric framework for the Zermelo navigation problem with arbitrary winds and for the causal geometry of Lorentzian spacetimes with a Killing vector field. Wind Finslerian structures encompass and extend Randers and Kropina metrics, leading to a comprehensive theory connecting Finsler geometry, Lorentz-Finsler geometry, and causality in mathematical relativity (Caponio et al., 2024, Javaloyes et al., 2017, Caponio et al., 2014).

1. Definition and Foundational Properties

Let $M$ be a smooth $n$-dimensional manifold. A wind Finslerian structure is specified by a smooth embedded hypersurface

$\Sigma \subset TM$

such that for each $p\in M$:

• $\Sigma_p = \Sigma \cap T_pM$ is a compact, connected, strongly convex $(n-1)$-sphere in $T_pM$ (not necessarily enclosing the origin).
• $\Sigma$ is transverse to the fibers, i.e.,

$$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$

Defining $B_p$ as the region bounded by $n$0 and $n$1 as the associated conic domain

$n$2

one obtains an open, conic, smooth bundle $n$3.

There exists a unique smooth, positively 1-homogeneous function

$n$4

whose restriction to each $n$5 is a conic Minkowski norm with strong convexity

$n$6

positive definite for all $n$7. The indicatrix at $n$8 is

$n$9

In the general setting, two pseudo-Finsler metrics emerge:

• $\Sigma \subset TM$0, with indicatrix $\Sigma \subset TM$1 (convex at $\Sigma \subset TM$2)
• $\Sigma \subset TM$3, Lorentz-Finsler with indicatrix $\Sigma \subset TM$4 (concave at $\Sigma \subset TM$5) These satisfy $\Sigma \subset TM$6 on the appropriate open cones, with boundary agreement on $\Sigma \subset TM$7 (Caponio et al., 2014).

2. Relationships with Cone Structures and Cone-Killing Fields

A cone structure on $\Sigma \subset TM$8 is an embedded hypersurface

$\Sigma \subset TM$9

with each fiber $p\in M$0 a smooth, connected, conic, strongly convex hypersurface satisfying

• $p\in M$1 for $p\in M$2, $p\in M$3
• transversality as above

A "cone-Killing" vector field $p\in M$4 is a vector field whose local flow leaves $p\in M$5 invariant:

$p\in M$6

If $p\in M$7 admits a complete cone-Killing field $p\in M$8 transverse to a spacelike hypersurface $p\in M$9, the flow yields $\Sigma_p = \Sigma \cap T_pM$0 with a canonical 1-form $\Sigma_p = \Sigma \cap T_pM$1 such that $\Sigma_p = \Sigma \cap T_pM$2, $\Sigma_p = \Sigma \cap T_pM$3. The fiberwise indicatrix $\Sigma_p = \Sigma \cap T_pM$4 for $\Sigma_p = \Sigma \cap T_pM$5 is

$\Sigma_p = \Sigma \cap T_pM$6

The associated wind Finslerian metric $\Sigma_p = \Sigma \cap T_pM$7 is extracted from this structure, and $\Sigma_p = \Sigma \cap T_pM$8 collectively determines $\Sigma_p = \Sigma \cap T_pM$9 (Caponio et al., 2024).

This construction encapsulates the Finslerian encoding of causal cones in Lorentzian geometry and is essential in understanding causality in standard spacetimes with Killing vector fields (Caponio et al., 2024, Caponio et al., 2014).

3. Wind Riemannian Structures, Randers, and Kropina Metrics

A major special case is the wind Riemannian structure, where each indicatrix $(n-1)$0 is an ellipsoid. This reduces the theory to Zermelo data:

• $(n-1)$1: a Riemannian metric on $(n-1)$2
• $(n-1)$3: a vector field ("wind")

The structure is given fiberwise by

$(n-1)$4

The conic Finsler metric is

$(n-1)$5

with $(n-1)$6.

The regime is classified as:

• Mild wind ($(n-1)$7): $(n-1)$8 is a Randers metric, $(n-1)$9
• Critical wind ($T_pM$0): $T_pM$1 is the Kropina norm, $T_pM$2, defined on the open half-space $T_pM$3
• Strong wind ($T_pM$4): $T_pM$5 is conic-Finsler, $T_pM$6 is Lorentz-Finsler, both defined only on the appropriate cones

Wind Finslerian structures thus generalize classical Finsler, Randers, and Kropina metrics and extend the range of geometric modeling to singular and unbounded wind domains (Javaloyes et al., 2017, Caponio et al., 2014, Javaloyes et al., 2017).

4. Causality: The Finslerian Causal Ladder

In the presence of a cone-Killing field $T_pM$7 and a cone structure $T_pM$8, the causal structure on $T_pM$9 can be classified via metric properties of the induced wind Finslerian structure $\Sigma$0 on a hypersurface $\Sigma$1. There is an explicit dictionary between traditional Lorentzian causal hierarchy and Finslerian notions:

Causality property	Finslerian criterion
Globally hyperbolic	Closed forward/backward $\Sigma$2-balls have compact intersection
Causally simple	$\Sigma$3 convex: geodesic joins any $\Sigma$4 with finite $\Sigma$5
Causally continuous	Symmetry of closure of wind balls $\Sigma$6
Cauchy hypersurface	Completeness: all closed $\Sigma$7-balls are compact

Here, the Finslerian separation

$\Sigma$8

controls the optimal time-separation in the associated spacetime $\Sigma$9 (Caponio et al., 2024, Caponio et al., 2014).

In the strong wind regime, pairs $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$0 are used; the causal types are reflected by domains of definiteness of these norms, and the corresponding wind balls and "c-balls" give the relevant compactness and completeness conditions.

5. Geodesics, Extremals, and Zermelo Navigation

The geodesics of a wind Finslerian structure stem from the Euler–Lagrange equations for $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$1 (or $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$2),

$$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$3

or can equivalently be described as solutions of the drifted Riemannian equation

$$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$4

when Zermelo data is present (Javaloyes et al., 2017). The corresponding geodesics divide into:

1. Minimizing (unit $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$5-geodesics)
2. Maximizing (unit $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$6-geodesics in strong wind)
3. "Abnormal" (boundary) geodesics, corresponding to pregeodesics of the degenerate directions

The wind Finslerian approach subsumes the generalized Zermelo navigation problem, including situations with strong or critical wind, and yields sharp existence results for minimizers and maximizers (length/minimizing and maximizing extremals) (Caponio et al., 2014).

6. Lorentzian Correspondence and SSTK Spacetimes

A canonical Lorentzian structure associated to a wind Finslerian structure is given by the standard spacetime with space-transverse Killing vector (SSTK):

$$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$7

with $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$8. The causal cones project to indicatrices of the wind Finslerian structure, and geodesic completeness of $$T_v(T_pM) + T_v\Sigma = T_v(TM)\quad\text{for all } v\in\Sigma_p.$$9 is equivalent to global hyperbolicity of $B_p$0 (Javaloyes et al., 2017, Caponio et al., 2014, Javaloyes et al., 2017).

This correspondence provides a powerful toolkit for relating Finslerian convexity/completeness issues to causality and Cauchy hypersurface criteria in Lorentzian geometry. SSTK spacetimes encompass standard stationary, Kropina, and strong-wind (ergosphere) metrics within a unified framework.

7. Completeness, Hopf–Rinow Theorem, and Global Structure

Geodesic completeness for wind Finslerian structures employs an analogue of the Hopf–Rinow theorem:

• $B_p$1 is geodesically complete $B_p$2 $B_p$3 is forward (or backward) Cauchy complete, every closed $B_p$4-bounded subset is compact, and the exponential map is defined on all of $B_p$5 (Javaloyes et al., 2017).
• For wind Riemannian structures, the completeness of $B_p$6 is implied by completeness of a suitable auxiliary metric, e.g., $B_p$7.

Completeness of $B_p$8 equivalently characterizes Cauchy hypersurfaces in the corresponding SSTK spacetime. This yields an operational bridge between Finsler geometry and the causal theory of spacetimes (Javaloyes et al., 2017).

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Wind Finslerian structures constitute a broad, rigorous framework capturing the interplay between generalized Finsler geometries, optimal navigation under arbitrary winds, and advanced causal properties in Lorentzian geometry, connecting these fields through deep geometric and analytic correspondences.