Mean Cartan Torsion in Finsler Geometry

• Mean Cartan torsion is a Finsler-geometric invariant defined as the contraction of the Cartan torsion with the inverse fundamental tensor, serving as a measure of non-Riemannianity.
• It plays a critical role in the structural analysis of (α,β)-metrics, where it governs the semi-C-reducible decomposition and links to bounded Cartan torsion in various metric families.
• The boundedness of mean Cartan torsion is pivotal for addressing embedding problems and understanding rigidity in Minkowskian products and other Finsler metric classifications.

to=arxiv_search.search 老时时彩 天天中彩票不能 code once {"query":"all:\"mean Cartan torsion\" OR ti:\"Cartan torsion\" Finsler", "max_results": 10, "sort_by": "relevance"}【อ่านข้อความเต็มanalysis to=arxiv_search.search code 大奖彩票站 ? Let's wait. to=arxiv_search.search 天天彩 to=arxiv_search.search 񟢀code ერთხელ {"query":"mean Cartan torsion Finsler", "max_results": 10} to=arxiv_search.search 】【。】【”】【code ერთხელ {"query":"(Gangopadhyay, 5 Aug 2025)", "max_results": 5} Mean Cartan torsion is a Finsler-geometric invariant obtained by contracting the Cartan torsion with the inverse fundamental tensor. For a Finsler metric $F$, it is written in components as

$I_i=g^{jk}C_{ijk},$

or, equivalently, as the linear form $I_y(u)=I_i(y)u^i$. In the literature represented here, mean Cartan torsion appears as a contracted measure of non-Riemannianity, as a structural variable in the semi-$C$-reducible theory of $(\alpha,\beta)$-metrics, and as a rigidly constrained object in Minkowskian products of Finsler manifolds (Tayebi et al., 2013, Gangopadhyay, 5 Aug 2025).

1. Definition within Finsler geometry

For a Finsler metric $F$ on a manifold $M$, the fundamental tensor at $y\in T_xM\setminus\{0\}$ is defined by

$$g_y(u,v) = \frac{1}{2}\frac{\partial^2}{\partial s\,\partial t} \Big[F^2(y+su+tv)\Big]_{s=t=0}.$$

The Cartan torsion is then obtained by differentiating $g_y$ in the fiber direction: $I_i=g^{jk}C_{ijk},$0 with local-coordinate expression

$I_i=g^{jk}C_{ijk},$1

A second formulation used in product geometry writes

$I_i=g^{jk}C_{ijk},$2

again giving

$I_i=g^{jk}C_{ijk},$3

The mean Cartan torsion is the contraction

$I_i=g^{jk}C_{ijk},$4

This is the standard definition adopted in both the bounded-torsion study of $I_i=g^{jk}C_{ijk},$5-metrics and the analysis of Minkowskian products (Tayebi et al., 2013, Gangopadhyay, 5 Aug 2025).

2. Vanishing criteria and diagnostic role

The full Cartan torsion satisfies

$I_i=g^{jk}C_{ijk},$6

while the mean Cartan torsion satisfies Deicke’s theorem,

$I_i=g^{jk}C_{ijk},$7

Accordingly, both tensors detect deviation from the Riemannian case, with $I_i=g^{jk}C_{ijk},$8 representing the contracted form of that deviation (Tayebi et al., 2013).

The same body of work emphasizes that the norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. In particular, if a Finsler manifold has unbounded Cartan torsion, then it cannot be isometrically imbedded into any Minkowski space. This places boundedness questions for Cartan torsion, and by extension questions involving its contraction $I_i=g^{jk}C_{ijk},$9, at the center of a geometric obstruction theory (Tayebi et al., 2013).

The norm used for Cartan torsion is

$I_y(u)=I_i(y)u^i$0

and, in local form,

$I_y(u)=I_i(y)u^i$1

This suggests that mean Cartan torsion is most naturally studied not in isolation, but as part of a hierarchy of fiberwise non-Riemannian invariants controlled by the same fundamental tensor.

3. Structural relation to Cartan torsion in $I_y(u)=I_i(y)u^i$2-metrics

For a non-Riemannian $I_y(u)=I_i(y)u^i$3-metric $I_y(u)=I_i(y)u^i$4, $I_y(u)=I_i(y)u^i$5, in dimension $I_y(u)=I_i(y)u^i$6, the norms of Cartan torsion and mean Cartan torsion are related by

$I_y(u)=I_i(y)u^i$7

where $I_y(u)=I_i(y)u^i$8, $I_y(u)=I_i(y)u^i$9, and

$C$0

The same framework gives the semi-$C$1-reducible decomposition

$C$2

with $C$3 the angular metric (Tayebi et al., 2013).

These formulas are the main conceptual bridge between Cartan torsion and mean Cartan torsion in the paper. They show that, for $C$4-metrics, the Cartan tensor is tightly controlled by the mean Cartan torsion. In this setting, $C$5 is not merely a trace-type auxiliary field; it is one of the parameters organizing the full cubic tensorial structure.

4. Bounded-torsion families and related subclasses

The boundedness results established for specific $C$6-metrics concern Cartan torsion directly, but they are especially relevant in view of the structural dependence on $C$7 described above. Two principal metric families are singled out (Tayebi et al., 2013):

Metric family	Conditions	Conclusion
$C$8	$C$9	$(\alpha,\beta)$0 has bounded Cartan torsion
$(\alpha,\beta)$1	$(\alpha,\beta)$2	$(\alpha,\beta)$3 has bounded Cartan torsion

The first family is the generalized Randers metric,

$(\alpha,\beta)$4

which includes the classical Randers metric when $(\alpha,\beta)$5. The second is the extended Berwald-type metric,

$(\alpha,\beta)$6

A separate theme is the existence of subclasses for which the bound of Cartan torsion is independent of $(\alpha,\beta)$7. One theorem identifies solutions of the form

$(\alpha,\beta)$8

or

$(\alpha,\beta)$9

and another gives a family

$F$0

For these metrics, the norm of Cartan torsion is independent of $F$1 (Tayebi et al., 2013).

Within complete $F$2-quadratic settings, the same paper cites Shen’s theorem and derives the implication

$F$3

A plausible implication is that mean Cartan torsion inherits much of its interest from this boundedness program, because its contraction data enters the semi-$F$4-reducible control of $F$5.

5. Mean Cartan torsion in Minkowskian products

A more recent line of work studies mean Cartan torsion for the Minkowskian product of two Finsler manifolds $F$6 and $F$7. On the product manifold

$F$8

with

$F$9

the product metric is defined by

$M$0

In the special linear case

$M$1

the product is called Euclidean Minkowskian product (Gangopadhyay, 5 Aug 2025).

For the mean Cartan torsion of this product, the explicit decomposition is

$M$2

where $M$3 and $M$4 are the mean Cartan torsions of the factor metrics and $M$5 are correction terms depending on $M$6, its derivatives, $M$7, and the dimensions $M$8. The essential conclusion is rigid:

$M$9

if and only if $y\in T_xM\setminus\{0\}$0 is linear in $y\in T_xM\setminus\{0\}$1 and $y\in T_xM\setminus\{0\}$2, that is,

$y\in T_xM\setminus\{0\}$3

Thus, the mean Cartan torsion of the product decomposes as the mean Cartan torsions of the factors if and only if the product is Euclidean. The same paper emphasizes the contrast with the full Cartan torsion norm: for any Minkowskian product of Finsler metrics, the norm of the Cartan torsion admits an additive decomposition in terms of the Cartan torsions of the component metrics, whereas a similar decomposition for the mean Cartan torsion holds only when the Minkowskian product is Euclidean (Gangopadhyay, 5 Aug 2025).

In the Euclidean case, the squared norm of the mean Cartan torsion becomes

$y\in T_xM\setminus\{0\}$4

with factor norms

$y\in T_xM\setminus\{0\}$5

In that case, boundedness of the norm of the mean Cartan torsion is equivalent to boundedness of the corresponding factor torsions, and a corollary states that the mean Cartan torsion of the Euclidean Minkowskian product of two Randers metrics is bounded (Gangopadhyay, 5 Aug 2025).

6. Terminological boundaries and non-Finsler uses of “Cartan torsion”

Mean Cartan torsion in the sense above belongs to Finsler geometry. It should not be conflated with torsion in Riemann–Cartan or Einstein–Cartan geometry.

In the Riemann–Cartan surface theory of “Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry,” the torsion under discussion is the torsion of a metric-compatible connection,

$y\in T_xM\setminus\{0\}$6

and the paper’s central invariant is the complex-valued quantity

$y\in T_xM\setminus\{0\}$7

That paper explicitly states that no direct connection is made to mean Cartan torsion from Finsler geometry or to any notion of “Cartan torsion” in that sense (Lee, 18 Feb 2025).

Likewise, “Kinematics of Einstein-Cartan universes” uses Cartan terminology for spacetime torsion and contortion,

$y\in T_xM\setminus\{0\}$8

within Einstein-Cartan gravity. It explicitly does not discuss mean Cartan torsion in the Finsler-geometric sense; the relevant objects there are the spacetime torsion tensor, the torsion vector $y\in T_xM\setminus\{0\}$9, and the associated kinematics of cosmological congruences (Pasmatsiou et al., 2016).

This distinction is substantive rather than merely linguistic. In Finsler geometry, mean Cartan torsion is the contraction $$g_y(u,v) = \frac{1}{2}\frac{\partial^2}{\partial s\,\partial t} \Big[F^2(y+su+tv)\Big]_{s=t=0}.$$0 of the fiberwise Cartan tensor. In Riemann–Cartan and Einstein–Cartan geometry, “torsion” refers instead to the antisymmetric part of an affine connection. The shared historical association with Cartan does not identify the objects.