Canonical selection among, and degenerate extension of, Chern's family of Euclidean connections in Finsler spaces

Chern's 1943 PNAS note proved, by Cartan's method of equivalence, that a regular Finsler space carries an infinite family of admissible Euclidean connections (invariant frame connections on the bundle of contact elements), parameterized by tensor fields, with Cartan's connection appearing as one member. The parameterization of the admissible family is complete in the note itself (its equation (31)); there is no separate classification problem. The open core has two linked parts: (A) whether an intrinsic, coordinate-free criterion can single out a unique preferred connection from the family -- later literature supplies axiomatic characterizations and uniqueness theorems for the standard members (Berwald, Cartan, Chern-Rund, Hashiguchi, including metrical axiom systems from which Chern's and Cartan's connections and the Hessian nature of the metric can be derived), but different axioms select different connections and no unique canonical choice, nor an impossibility result, is known; and (B) the extension of the Euclidean-connection and invariant theory to Finsler functions whose velocity Hessian is degenerate (the non-regular case), which remains essentially untouched -- existing pseudo-Finsler treatments preserve non-degeneracy and only change signature, while singular-metric work provides at most nonlinear connections and special curvature computations.

Background

A Finsler space generalizes a Riemannian manifold by replacing the quadratic length element with a variational integral $s=\int F(x^i,\dot x^i)\,dt$ whose fundamental function $F$ is positively homogeneous of degree one in the velocity argument ($F(x,\lambda y)=\lambda F(x,y)$ for $\lambda>0$); Riemannian geometry is the special case $F=\sqrt{g_{ij}(x)y^iy^j}$. A contact element is a pair $(x,y)$ consisting of a point of the manifold and a nonzero tangent vector at it, i.e. a point of the slit tangent bundle. The central parallelism problem of Finsler geometry, discussed from the 1920s on, was to generalize the Levi-Civita connection of Riemannian geometry (the unique torsion-free connection compatible with the metric) to a Finsler space; J. L. Synge, J. H. Taylor, L. Berwald and E. Cartan each proposed a construction. In E. Cartan's approach, a Euclidean connection attaches to every contact element a tangent Euclidean space with an orthonormal frame $(M,e_1,\dots,e_n)$ and specifies the infinitesimal displacement of this frame between neighboring contact elements by equations of the form $dM=\pi^ie_i$, $de_i=\pi^j{}_ie_j$ (Cartan's method of moving frames). Cartan's own Finsler connection appeared in his 1934 monograph 'Les espaces de Finsler'.

In this note, Chern applied the method of equivalence (Cartan's algorithm that builds invariant coframes on a fibered structure and normalizes torsion by absorbing it into the connection forms) to this problem. He worked on the space of variables $(x^i,y^i,v_{i\alpha})$, where the $v_{i\alpha}$ are $n(n-1)$ auxiliary frame coefficients with $v_{i\alpha}y^i=0$, $\alpha=1,\dots,n-1$, and imposed the regularity assumption: the velocity Hessian $\bigl(\partial^2F/\partial y^i\partial y^j\bigr)$, which always annihilates the radial direction $y$, has maximal rank $n-1$ (in the calculus of variations this is the regularity, or strong convexity, of the length functional). Under this assumption he derived the family of all invariant frame connections: the connection forms are determined up to a tensor $\gamma_{ij\alpha}$ subject to algebraic symmetry conditions (his eq. (29)) and a differential invariance condition (his eq. (30)), and he verified that (30) is equivalent to $\gamma$ being of the form $G_{ijk}u^iu^ju^k$ where the $G_{ijk}$ are functions of $(x,y)$ only (his eq. (31)); $u$ denotes the normalized frame covectors. Cartan's connection is the member $\gamma_{pq\alpha}=-\tfrac12H_{pq\alpha}$ (eq. (33)), where $H$ is built from the second velocity derivatives of $F$ and vanishes exactly when the space is Riemannian. Chern concluded that there are 'as many Euclidean connections in the space as there are invariants satisfying our conditions', that along any curve of contact elements the attached Euclidean spaces develop onto one and the same Euclidean space (Fermi's transport theorem generalized), and that the family of connections together with the curvature invariants determines the Finsler space up to point transformation.

The modern theory recast this picture in the language of connections on the pullback tangent bundle $\pi^*TM$ over the slit tangent bundle. A Finsler connection decomposes into a nonlinear connection (a horizontal distribution, in practice Berwald's), horizontal connection coefficients $\Gamma^i{}_{jk}$ (equivalently the spray, the velocity-quadratic vector field on the slit tangent bundle whose integral curves project onto the geodesics), and vertical coefficients $V^i{}_{jk}$; the four most-used connections (Berwald, Cartan, Chern-Rund, Hashiguchi) are the four noteworthy combinations of these choices, and they differ precisely by the Cartan torsion $C_{ijk}=\tfrac12\partial^3L/\partial y^i\partial y^j\partial y^k$ (the third velocity derivative of the quadratic Lagrangian $L=\tfrac12F^2$, which vanishes exactly in the Riemannian case). The connection now called Chern's (historically found independently by H. Rund, hence 'Chern-Rund') emerged from Chern's fuller 1948 development of the same equivalence method. Standard references for this framework are Matsumoto's monograph on the theory of Finsler connections and the graduate text of Bao, Chern and Shen.

Problem Statement

Let $F(x,y)$ be a positively one-homogeneous fundamental function on the slit tangent bundle of an $n$-dimensional manifold. Chern (1943), assuming regularity (the velocity Hessian $\bigl(\partial^2F/\partial y^i\partial y^j\bigr)$ has rank $n-1$), proved that the admissible Euclidean connections - invariant frame connections on the bundle of contact elements $(x,y)$ satisfying his symmetry and differential invariance conditions - form a family parameterized by tensor fields $G_{ijk}(x,y)$ (his equation (31)), of which Cartan's connection ($\gamma=-\tfrac12H$) is one member. The parameterization itself is complete; what remains unresolved, and is asked here, are the two linked questions that the note leaves open:

(A) Canonical selection. Determine whether an intrinsic, coordinate-free criterion - a naturality criterion, meaning one equivariant under point transformations of the Finsler space and under the structure group of the slit tangent bundle, hence formulated without reference to coordinates or to a preselected connection - singles out a unique preferred connection from this family, i.e. from the admissible Euclidean connections parameterized by the tensor fields $G_{ijk}(x,y)$ (Chern's equation (31)). Here 'criterion' means a system of natural conditions that a member of the family may or may not satisfy - of the same kind as the torsion and metric-compatibility conditions that characterize the standard connections - and not an explicit formula prescribing the parameter $G_{ijk}(x,y)$. The question is restricted to the members of this family other than its four standard connections - Berwald's, Cartan's, Chern-Rund's and Hashiguchi's (Cartan's being $\gamma=-\tfrac12H$), each of which is already singled out as the unique solution of its own torsion-plus-metricity axioms, with metrical axiom systems already selecting Chern's and Cartan's. A satisfactory answer is either (i) a naturality or compatibility axiom system, stated intrinsically, whose unique solution in the family is a single member other than those four standard connections, or (ii) a rigorous impossibility result showing that no criterion of this kind singles out a unique member of the family other than those four standard connections.

(B) Degenerate extension. Construct, or prove obstructed, an analogous theory of Euclidean connections and of the associated invariants for fundamental functions that fail the regularity condition, i.e. whose velocity Hessian has rank less than $n-1$. The theory should specify the replacement for the Euclidean frames (which rely on non-degeneracy of the fundamental tensor), define the infinitesimal displacement of the attached spaces, reproduce the invariance conditions and the development property along curves of contact elements in a form that reduces to Chern's construction in the regular case, and say which invariants classify such spaces up to point transformation.

Both strands serve the single objective left by the 1943 note: completing the theory of Euclidean connections of Finsler spaces - the family, its distinguished members, and its domain of validity.

References

1. S.-S. Chern, 'On the Euclidean Connections in a Finsler Space', Proceedings of the National Academy of Sciences 29(1), 33-37 (1943), DOI 10.1073/pnas.29.1.33, https://doi.org/10.1073/pnas.29.1.33
2. S.-S. Chern, 'Local Equivalence and Euclidean Connections in Finsler Spaces', Science Reports of National Tsing Hua University, Series A 5, 95-107 (1948); reprinted in S.-S. Chern, Selected Papers II, Springer (1989), pp. 194-212, chapter DOI 10.1007/978-1-4612-3546-0_20, https://doi.org/10.1007/978-1-4612-3546-0_20
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