---
title: Pseudo-Week Structure in Pseudo-Finsler
url: https://www.emergentmind.com/topics/pseudo-week-structure
type: topic
---

# Pseudo-Week Structure in Pseudo-Finsler

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{"query":"arXiv 1311.0199 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 [1311.0199].

## 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 \(F:T\to \mathbb{R}^+\) satisfying three conditions. First, for each \(p\in M\), the fiberwise domain
\[
T_p := T \cap T_pM
\]
is a nonempty open cone in \(T_pM\): if \(v\in T_p\) and \(t>0\), then \(tv\in T_p\), and \(T_p\) is open in the vector space \(T_pM\). Second, \(F\) is positively 1-homogeneous,
\[
F(tv)=tF(v), \qquad v\in T,\ t>0.
\]
Third, the fundamental tensor
\[
g_v=\frac{1}{2}\frac{\partial^2}{\partial y^i\partial y^j}(F^2)\big|_v\,dy^i\otimes dy^j
\]
is nondegenerate at every \(v\in T\). Equivalently, the matrix \(\big(g_v(\partial_{x^i},\partial_{x^j})\big)\) is invertible for each \(v\) [1311.0199].

Because nondegeneracy is required on the full conical domain, the index of \(g_v\), namely the number of negative eigenvalues, is constant on \(T\). This constant is the index of the pseudo-Finsler structure. The framework therefore includes both definite and indefinite geometries. When \(T=T_0M\) and each \(g_v\) is positive definite, one recovers a classical Finsler metric. When \(g_v\) is independent of \(v\) and depends only on \(p=\pi(v)\), 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 \(T_p\) 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 \(v\in TM\), the vertical subspace is
\[
\mathrm{Ver}_v=\ker(d\pi_v)\subset T_v(TM),
\]
and there is a canonical identification
\[
i_v:T_{\pi(v)}M\longrightarrow \mathrm{Ver}_v.
\]
Using these maps, one defines the quasi-tangent structure \(J\) on \(TM\) by
\[
J(X)=i_v(d\pi(X)), \qquad X\in T_v(TM).
\]
If \(f:M\to M\) is a \(C^2\) diffeomorphism, then its tangent map preserves this structure:
\[
(df)^*J=J.
\]
Accordingly, \(df\) preserves the vertical distribution [1311.0199].

A pseudo-Finsler structure also has a canonical geodesic spray \(S\in \mathfrak{X}(T)\), characterized in part by
\[
J(S)=C, \qquad [C,S]=S,
\]
where \(C\) is the Liouville vector field,
\[
C_v=i_v(v).
\]
From the Lie derivative of \(J\) along \(S\),
\[
T^S=-\mathcal{L}_S(J),
\]
Grifone’s theory yields
\[
\ker(T^S+\mathrm{Id})=\mathrm{Ver}, \qquad (T^S)^2=\mathrm{Id}.
\]
This produces the horizontal distribution
\[
\mathrm{Hor}^S=\ker(T^S-\mathrm{Id}),
\]
and a direct sum decomposition
\[
T(T)\big|_T=\mathrm{Ver}\oplus \mathrm{Hor}^S.
\]
For each \(v\in T\), the map \(d\pi|_{\mathrm{Hor}^S_v}\) is an isomorphism onto \(T_{\pi(v)}M\).

The associated Sasaki-type metric \(g^F\) is then defined on the manifold \(T\), viewed as an open subset of \(TM\). On the vertical subspace, \(g_v\) is pushed forward via \(i_v\); on the horizontal subspace, \(g_v\) is pulled back via \(d\pi|_{\mathrm{Hor}^S_v}\); and horizontal and vertical directions are declared orthogonal. If \(X,Y\in T_v(T)\) are decomposed into horizontal and vertical parts, then
\[
g^F_v(X^{\mathrm{hor}},Y^{\mathrm{hor}})=g_v\big(d\pi(X^{\mathrm{hor}}),d\pi(Y^{\mathrm{hor}})\big),
\]
\[
g^F_v(X^{\mathrm{ver}},Y^{\mathrm{ver}})=g_v\big(i_v^{-1}(X^{\mathrm{ver}}),i_v^{-1}(Y^{\mathrm{ver}})\big),
\]
and mixed terms vanish. The result is a smooth nondegenerate symmetric \((0,2)\)-tensor on \(T\) with
\[
\mathrm{index}(g^F)=2k,
\]
where \(k\) 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 \(M\) to a pseudo-Riemannian structure on \(T\), 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 \((M,T,F)\) is a \(C^2\) diffeomorphism \(f:M\to M\) such that
\[
df(T)=T
\]
and
\[
F\circ df = F.
\]
Equivalently, the fundamental tensor is preserved:
\[
g_{df(v)}(df(X),df(Y))=g_v(X,Y)
\]
for all \(v\in T\) and all \(X,Y\in T_{\pi(v)}M\). The full isometry group is denoted \(\mathrm{Iso}(M,T,F)\), and it is a group under composition [1311.0199].

The central proposition identifies such isometries with isometries of the Sasaki-type metric after tangent lifting. If \(f\) is a pseudo-Finsler isometry, then
\[
df:T\to T
\]
is an isometry of the pseudo-Riemannian manifold \((T,g^F)\). The mechanism is structural rather than ad hoc. Preservation of \(J\) forces preservation of the vertical distribution. Preservation of the pseudo-Finsler data implies preservation of geodesics and hence of the geodesic spray \(S\), which in turn preserves \(\mathrm{Hor}^S\). Since \(g^F\) is defined fiberwise from \(g_v\) on the horizontal and vertical components, preservation of the fundamental tensor yields preservation of \(g^F\).

There is also a converse statement: if \(f:M\to M\) is \(C^2\), satisfies \(df(T)=T\), and the tangent map \(df|_T:T\to T\) is an isometry of \((T,g^F)\), then \(f\) is a diffeomorphism of \(M\) and an isometry of \((M,T,F)\). Thus the assignment
\[
\Phi:\mathrm{Iso}(M,T,F)\ni f\longmapsto df\in \mathrm{Iso}(T,g^F)
\]
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 \((M,F)\), endowed with the \(C^1\)-topology, is a Lie transformation group of \(M\). More explicitly, \(\mathrm{Iso}(M,T,F)\) is a finite-dimensional Lie group, and the natural action
\[
\mathrm{Iso}(M,T,F)\times M\to M,\qquad (f,p)\mapsto f(p)
\]
is smooth [1311.0199].

The proof proceeds by embedding \(\mathrm{Iso}(M,T,F)\) into \(\mathrm{Iso}(T,g^F)\) via \(\Phi(f)=df\). Since pseudo-Riemannian isometry groups are Lie groups by the general theory of \(G\)-structures and transformation groups, \(\mathrm{Iso}(T,g^F)\) is a finite-dimensional Lie group whose action on \(T\) is smooth. The key additional step is a closedness result: the image of \(\Phi\) is closed in the \(C^1\)-topology. If a sequence \(df_n\) converges in \(C^1\) to a \(C^1\) diffeomorphism \(\Psi:T\to T\), local sections of \(\pi:T\to M\) allow one to reconstruct a limiting \(C^1\) map \(f_0:M\to M\), show that \(\Psi=df_0\), and then conclude from the converse statement above that \(f_0\) is a pseudo-Finsler isometry.

Because a closed subgroup of a Lie group is a Lie subgroup, \(\Phi(\mathrm{Iso}(M,T,F))\) 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 \(T\) to a smooth action on \(M\), one uses a general lemma about a smooth surjective submersion \(q:E\to B\): if a Lie transformation group of \(E\) preserves fibers of \(q\), it induces a Lie transformation group on \(B\). Applying this to
\[
E=T,\qquad B=M,\qquad q=\pi,
\]
and the action \(v\mapsto df(v)\), one obtains smoothness of the base action.

A corollary strengthens the rigidity statement: every pseudo-Finsler isometry is a \(C^\infty\)-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 \(O(p,q)\)-structure, which has finite order, and the \(C^1\)-topology becomes the natural choice [1311.0199].

For ordinary Finsler metrics, the paper recalls a reduction to the Riemannian case using an averaged metric. Given the indicatrix
\[
E_p=\{u\in T_pM:F(u)=1\},
\]
one defines
\[
h_F(v,w)=\int_{E_p} g_u(v,w)\,d\mu_p(u),
\]
where \(d\mu_p\) is the volume induced by \(g_u\). Finsler isometries preserve \(h_F\), so \(\mathrm{Iso}(M,F)\) embeds as a closed subgroup of \(\mathrm{Iso}(M,h_F)\), whence it is a Lie group.

The pseudo-Finsler case differs in two decisive ways: there is no metric-space structure, and the associated \(G\)-structure has infinite order. The paper therefore cannot use either the classical Riemannian route or the finite-order \(G\)-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 \(h_F\) | First-jet determination |
| Pseudo-Finsler | Sasaki pseudo-Riemannian metric \(g^F\) on \(T\) | 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 \(F\) is only defined on \(T\subset T_0M\), not on all nonzero tangent vectors. Even if \(g_v\) were positive definite on \(T\), there is no reason to obtain a natural distance function on all of \(M\). Hence classical metric arguments are unavailable. The paper also emphasizes that \(g^F\) is defined only on the conical domain \(T\), not on all of \(TM\) [1311.0199].

The indefinite character of the fundamental tensor introduces additional obstructions. The indicatrix \(E_p\cap T_p\) 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 \(\mathfrak{iso}(M,T,F)\) consists of vector fields whose flows are 1-parameter groups of pseudo-Finsler isometries. Via the embedding \(f\mapsto df\), this Lie algebra may be viewed as a subalgebra of the Lie algebra of Killing vector fields on \((T,g^F)\) that are projectable to \(M\) 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 \(M\), with smooth elements and a precise jet-determination property [1311.0199].

Source: https://www.emergentmind.com/topics/pseudo-week-structure