Pseudo-Week Structure in Pseudo-Finsler

• Pseudo-Week Structure is a framework for pseudo-Finsler isometries defined on conical subsets with positively 1-homogeneous functions.
• It leverages a Sasaki-type metric to convert pseudo-Finsler geometry into a pseudo-Riemannian structure, enabling Lie group analysis.
• This approach shows that the isometry group is finite-dimensional and determined by second-jet data, ensuring smooth and rigid actions.

Searching arXiv for the specified paper and closely related references to ground the article in current bibliographic data. {"query":"arXiv (Torromé et al., 2013) On the Lie group structure of pseudo-Finsler isometries", "max_results": 5} {"query":"Deng Hou Finsler isometries Lie transformation group arXiv", "max_results": 5} {"query":"S. Deng Z. Hou isometries of Finsler spaces Lie transformation groups arXiv", "max_results": 10} In the context of pseudo-Finsler geometry, “Pseudo-Week Structure” is plausibly interpreted as the structural theory of pseudo-Finsler isometries, specifically the result that the isometry group of a pseudo-Finsler conical metric carries a finite-dimensional Lie group structure and acts smoothly on the base manifold. The central framework is developed for a connected smooth manifold $M$, an open conical subset $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$, and a positively 1-homogeneous smooth function $F:T\to \mathbb{R}^+$ whose fundamental tensor is nondegenerate on every allowed tangent direction. The main theorem establishes that $\mathrm{Iso}(M,T,F)$, endowed with the $C^1$-topology, is a Lie transformation group of $M$, and does so by passing from the pseudo-Finsler data on $M$ to an associated pseudo-Riemannian Sasaki-type metric $g^F$ on the conical domain $T$ inside the tangent bundle (Torromé et al., 2013).

1. Pseudo-Finsler conical metrics and their basic data

A pseudo-Finsler conical metric is specified by an open subset $T \subset T_0M$ and a smooth function $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$0 satisfying three conditions. First, for each $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$1, the fiberwise domain

$$T \subset T_0M = TM \setminus \{\text{zero section}\}$$2

is a nonempty open cone in $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$3: if $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$4 and $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$5, then $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$6, and $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$7 is open in the vector space $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$8. Second, $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$9 is positively 1-homogeneous,

$F:T\to \mathbb{R}^+$0

Third, the fundamental tensor

$F:T\to \mathbb{R}^+$1

is nondegenerate at every $F:T\to \mathbb{R}^+$2. Equivalently, the matrix $F:T\to \mathbb{R}^+$3 is invertible for each $F:T\to \mathbb{R}^+$4 (Torromé et al., 2013).

Because nondegeneracy is required on the full conical domain, the index of $F:T\to \mathbb{R}^+$5, namely the number of negative eigenvalues, is constant on $F:T\to \mathbb{R}^+$6. This constant is the index of the pseudo-Finsler structure. The framework therefore includes both definite and indefinite geometries. When $F:T\to \mathbb{R}^+$7 and each $F:T\to \mathbb{R}^+$8 is positive definite, one recovers a classical Finsler metric. When $F:T\to \mathbb{R}^+$9 is independent of $\mathrm{Iso}(M,T,F)$0 and depends only on $\mathrm{Iso}(M,T,F)$1, one recovers a pseudo-Riemannian metric.

The conical restriction is essential. The geometry is not defined on all nonzero tangent vectors in general, but only on those lying in the admissible cone $\mathrm{Iso}(M,T,F)$2 at each point. This affects both geodesic theory and the automorphism theory. A plausible implication is that pseudo-Finsler geometry combines directional anisotropy with signature effects in a way that cannot be reduced to standard metric-space arguments.

2. Tangent-bundle structures and the Sasaki-type metric

The construction used to analyze pseudo-Finsler isometries begins on the tangent bundle. For $\mathrm{Iso}(M,T,F)$3, the vertical subspace is

$\mathrm{Iso}(M,T,F)$4

and there is a canonical identification

$\mathrm{Iso}(M,T,F)$5

Using these maps, one defines the quasi-tangent structure $\mathrm{Iso}(M,T,F)$6 on $\mathrm{Iso}(M,T,F)$7 by

$\mathrm{Iso}(M,T,F)$8

If $\mathrm{Iso}(M,T,F)$9 is a $C^1$0 diffeomorphism, then its tangent map preserves this structure: $C^1$1 Accordingly, $C^1$2 preserves the vertical distribution (Torromé et al., 2013).

A pseudo-Finsler structure also has a canonical geodesic spray $C^1$3, characterized in part by

$C^1$4

where $C^1$5 is the Liouville vector field,

$C^1$6

From the Lie derivative of $C^1$7 along $C^1$8,

$C^1$9

Grifone’s theory yields

$M$0

This produces the horizontal distribution

$M$1

and a direct sum decomposition

$M$2

For each $M$3, the map $M$4 is an isomorphism onto $M$5.

The associated Sasaki-type metric $M$6 is then defined on the manifold $M$7, viewed as an open subset of $M$8. On the vertical subspace, $M$9 is pushed forward via $M$0; on the horizontal subspace, $M$1 is pulled back via $M$2; and horizontal and vertical directions are declared orthogonal. If $M$3 are decomposed into horizontal and vertical parts, then

$M$4

$M$5

and mixed terms vanish. The result is a smooth nondegenerate symmetric $M$6-tensor on $M$7 with

$M$8

where $M$9 is the index of the original pseudo-Finsler structure.

This metric is the paper’s decisive technical device. It transfers the problem from a pseudo-Finsler structure on $g^F$0 to a pseudo-Riemannian structure on $g^F$1, where the Lie theory of isometry groups is already available.

3. Pseudo-Finsler isometries and their tangent lifts

An isometry of the pseudo-Finsler structure $g^F$2 is a $g^F$3 diffeomorphism $g^F$4 such that

$g^F$5

and

$g^F$6

Equivalently, the fundamental tensor is preserved: $g^F$7 for all $g^F$8 and all $g^F$9. The full isometry group is denoted $T$0, and it is a group under composition (Torromé et al., 2013).

The central proposition identifies such isometries with isometries of the Sasaki-type metric after tangent lifting. If $T$1 is a pseudo-Finsler isometry, then

$T$2

is an isometry of the pseudo-Riemannian manifold $T$3. The mechanism is structural rather than ad hoc. Preservation of $T$4 forces preservation of the vertical distribution. Preservation of the pseudo-Finsler data implies preservation of geodesics and hence of the geodesic spray $T$5, which in turn preserves $T$6. Since $T$7 is defined fiberwise from $T$8 on the horizontal and vertical components, preservation of the fundamental tensor yields preservation of $T$9.

There is also a converse statement: if $T \subset T_0M$0 is $T \subset T_0M$1, satisfies $T \subset T_0M$2, and the tangent map $T \subset T_0M$3 is an isometry of $T \subset T_0M$4, then $T \subset T_0M$5 is a diffeomorphism of $T \subset T_0M$6 and an isometry of $T \subset T_0M$7. Thus the assignment

$T \subset T_0M$8

is injective, and its image is exactly the collection of Sasaki isometries that arise as tangent maps of base diffeomorphisms preserving the conical domain and the fundamental function.

This injective lift is the formal bridge between pseudo-Finsler automorphisms and pseudo-Riemannian automorphisms.

4. Lie group structure and proof strategy

The main theorem states that the group of isometries of a pseudo-Finsler structure $T \subset T_0M$9, endowed with the $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$00-topology, is a Lie transformation group of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$01. More explicitly, $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$02 is a finite-dimensional Lie group, and the natural action

$$T \subset T_0M = TM \setminus \{\text{zero section}\}$$03

is smooth (Torromé et al., 2013).

The proof proceeds by embedding $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$04 into $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$05 via $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$06. Since pseudo-Riemannian isometry groups are Lie groups by the general theory of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$07-structures and transformation groups, $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$08 is a finite-dimensional Lie group whose action on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$09 is smooth. The key additional step is a closedness result: the image of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$10 is closed in the $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$11-topology. If a sequence $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$12 converges in $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$13 to a $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$14 diffeomorphism $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$15, local sections of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$16 allow one to reconstruct a limiting $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$17 map $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$18, show that $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$19, and then conclude from the converse statement above that $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$20 is a pseudo-Finsler isometry.

Because a closed subgroup of a Lie group is a Lie subgroup, $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$21 inherits a finite-dimensional Lie group structure. The original isometry group is then identified with this Lie subgroup. To pass from the smooth action on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$22 to a smooth action on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$23, one uses a general lemma about a smooth surjective submersion $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$24: if a Lie transformation group of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$25 preserves fibers of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$26, it induces a Lie transformation group on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$27. Applying this to

$$T \subset T_0M = TM \setminus \{\text{zero section}\}$$28

and the action $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$29, one obtains smoothness of the base action.

A corollary strengthens the rigidity statement: every pseudo-Finsler isometry is a $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$30-map, and it is completely determined by its second jet at any point. Thus if two isometries agree up to second order at one point, they agree everywhere. The paper explicitly contrasts this with the positive-definite Finsler case, where isometries are determined by their first jet.

5. Comparison with Riemannian, pseudo-Riemannian, and Finsler theories

The pseudo-Finsler result is situated between several established theories. In the Riemannian case, the Myers–Steenrod and Palais framework shows that isometry groups are Lie groups under the compact-open topology, and the metric-space structure is essential in the classical argument. In the pseudo-Riemannian case, one cannot use metric-space methods because the metric is indefinite; instead, one studies the automorphism group of the underlying $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$31-structure, which has finite order, and the $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$32-topology becomes the natural choice (Torromé et al., 2013).

For ordinary Finsler metrics, the paper recalls a reduction to the Riemannian case using an averaged metric. Given the indicatrix

$$T \subset T_0M = TM \setminus \{\text{zero section}\}$$33

one defines

$$T \subset T_0M = TM \setminus \{\text{zero section}\}$$34

where $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$35 is the volume induced by $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$36. Finsler isometries preserve $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$37, so $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$38 embeds as a closed subgroup of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$39, whence it is a Lie group.

The pseudo-Finsler case differs in two decisive ways: there is no metric-space structure, and the associated $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$40-structure has infinite order. The paper therefore cannot use either the classical Riemannian route or the finite-order $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$41-structure route directly. Nor can it use the Finsler averaging trick.

Geometry	Standard route	Order/jets
Riemannian	Metric-space methods and Lie theory	Smooth isometries
Finsler	Averaged Riemannian metric $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$42	First-jet determination
Pseudo-Finsler	Sasaki pseudo-Riemannian metric $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$43 on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$44	Second-jet determination

This comparison clarifies the role of the Sasaki lift. It is not merely an auxiliary construction; it is the replacement for both metric-space arguments and averaging.

6. Conical and indefinite features, rigidity, and consequences

The conical nature of the domain means that $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$45 is only defined on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$46, not on all nonzero tangent vectors. Even if $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$47 were positive definite on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$48, there is no reason to obtain a natural distance function on all of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$49. Hence classical metric arguments are unavailable. The paper also emphasizes that $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$50 is defined only on the conical domain $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$51, not on all of $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$52 (Torromé et al., 2013).

The indefinite character of the fundamental tensor introduces additional obstructions. The indicatrix $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$53 is never compact when the metric is indefinite, and sums or convex combinations of indefinite nondegenerate symmetric bilinear forms can become degenerate. This blocks the averaging construction that works in the positive-definite Finsler setting. A plausible implication is that pseudo-Finsler geometry requires a transformation-theoretic rather than an integral-geometric approach to automorphism groups.

Once the Lie group theorem is established, standard Lie-theoretic consequences follow. The Lie algebra $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$54 consists of vector fields whose flows are 1-parameter groups of pseudo-Finsler isometries. Via the embedding $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$55, this Lie algebra may be viewed as a subalgebra of the Lie algebra of Killing vector fields on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$56 that are projectable to $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$57 and compatible with the conical structure. The second-jet determination result implies a strong local rigidity: finite jet data at one point determine the global isometry.

The paper is primarily structural rather than example-driven. Its principal contribution is to show that the isometry group of a pseudo-Finsler conical metric is neither weakly organized nor pathological from the viewpoint of transformation groups. Despite the absence of a global distance, the conical restriction of allowed directions, and the indefinite signature of the fundamental tensor, the automorphism group is a finite-dimensional Lie group acting smoothly on $$T \subset T_0M = TM \setminus \{\text{zero section}\}$$58, with smooth elements and a precise jet-determination property (Torromé et al., 2013).