Partial Finsler Manifolds

• Partial Finsler manifolds are geometric structures where the norm is defined only on a conic subset of the tangent bundle, relaxing full positive-definiteness.
• They extend classical Finsler geometry by incorporating slit-based domains, anchored bundles, and leafwise methods to handle low-regularity and piecewise smooth cases.
• These structures support advanced analytical tools, including canonical connections, smoothing techniques, and rigidity results, to enrich the study of anisotropic spaces.

Searching arXiv for recent and foundational papers on partial Finsler manifolds and related low-regularity / conic Finsler geometry. Partial Finsler manifolds are Finsler-type geometric structures in which the norm is not required to be defined on the whole slit tangent bundle in the standard way. In the literature represented here, the term appears in several closely related senses. In Pelletier’s framework, a partial Finsler metric is a positively homogeneous, strongly convex function defined on a conic open subset of an anchored bundle, with geometry organized by the prolongation of a pre-Lie algebroid and induced leafwise on the foliation determined by the anchor (Pelletier, 2014). In the later terminology of almost and partial Finsler manifolds, a partial Finsler manifold is a triple $(M,S,F)$ where $S\subset TM$ is a closed conelike slit possibly containing nonzero directions and $F:TM-S\to (0,\infty)$ is smooth and positively homogeneous, while positive definiteness of the fundamental tensor is not required everywhere; the positive-definite subclass is called almost Finsler (Davis et al., 29 Aug 2025). Adjacent work on partially smooth metrics, conic Finsler manifolds, and $C^0$-Finsler structures shows that partiality can arise from restricted domains, low regularity, or piecewise smooth constructions, and that many classical Finsler tools can still be recovered or approximated in these broader settings (Matveev et al., 2016, Youssef et al., 2018, Fukuoka et al., 2019).

1. Definitions and basic models

A foundational version of partial Finsler geometry is formulated on an anchored bundle $(A,M,p)$. Fix a conic open subset $\mathcal M\subset A$ that does not contain the zero section and is stable under positive fiberwise scaling. A partial Finsler metric is then a continuous map

$F:\mathcal M\to [0,+\infty[$

such that $F$ is smooth on $\mathcal M$, satisfies $F(\lambda u)=\lambda F(u)$ for $S\subset TM$0, and has positive-definite fundamental quadratic form

$S\subset TM$1

for every $S\subset TM$2 (Pelletier, 2014). In this sense, the structure is “partial” because it is defined only on a conic domain in the fibers rather than on all nonzero directions of a vector bundle.

A later, explicitly manifold-based definition uses a slit $S\subset TM$3. A partial Finsler manifold is a triple

$S\subset TM$4

with $S\subset TM$5 a closed conelike subset containing the zero section and

$S\subset TM$6

smooth and homogeneous of degree $S\subset TM$7 in each tangent space, such that each fiber restriction is a partial Minkowski norm (Davis et al., 29 Aug 2025). In this framework, an almost Finsler manifold is a partial Finsler manifold whose fiberwise Hessian of $S\subset TM$8 is positive definite on $S\subset TM$9, and a standard Finsler manifold is recovered by taking the slit to be exactly the zero section (Davis et al., 29 Aug 2025).

The corresponding local model is a partial Minkowski space $F:TM-S\to (0,\infty)$0, where $F:TM-S\to (0,\infty)$1 is a finite-dimensional real vector space, $F:TM-S\to (0,\infty)$2 is closed and conelike, and $F:TM-S\to (0,\infty)$3 is positively homogeneous: $F:TM-S\to (0,\infty)$4 If the bilinear form

$F:TM-S\to (0,\infty)$5

is positive definite for every $F:TM-S\to (0,\infty)$6, one obtains an almost Minkowski space (Davis et al., 29 Aug 2025). This fiberwise picture clarifies the two main departures from standard Finsler geometry: the slit may contain nonzero scaling-invariant directions, and the Hessian need not stay positive definite on the whole domain.

A related but distinct regularity-based notion is $F:TM-S\to (0,\infty)$7-partial smoothness. A continuous Finsler metric $F:TM-S\to (0,\infty)$8 on $F:TM-S\to (0,\infty)$9 is $C^0$0-partially smooth if there exists a positively homogeneous, fiberwise bilipschitz map

$C^0$1

such that, for all $C^0$2,

$C^0$3

where

$C^0$4

This allows a metric to become regular only after a fiberwise change of variables (Matveev et al., 2016). The terminology differs from the slit-based usage, but both frameworks enlarge the standard Finsler category.

2. Fundamental tensors, slit geometry, and truncation

On a partial Finsler manifold in the slit-based sense, the standard Finsler tensors are defined on the pullback bundle over $C^0$5. With fiber coordinates $C^0$6, the fundamental tensor is

$C^0$7

the Cartan tensor is

$C^0$8

the Hilbert form is

$C^0$9

the angular metric is

$(A,M,p)$0

and the mean Cartan torsion is

$(A,M,p)$1

(Davis et al., 29 Aug 2025). Homogeneity yields the usual Euler identities, including

$(A,M,p)$2

(Davis et al., 29 Aug 2025).

In the anchored-bundle setting, the fundamental tensor is the vertical Riemannian metric $(A,M,p)$3, and the Cartan tensor is the symmetric trilinear form

$(A,M,p)$4

satisfying $(A,M,p)$5 (Pelletier, 2014). This reproduces the standard Finsler package on the allowed conic directions.

The slit geometry is itself part of the structure. For a partial Finsler manifold $(A,M,p)$6, the slit tangent bundle is $(A,M,p)$7, and the paper introducing almost and partial Finsler manifolds defines the extended slit

$(A,M,p)$8

The triple $(A,M,p)$9 is then called the truncation and is always an almost Finsler manifold (Davis et al., 29 Aug 2025). This construction separates mere smooth homogeneity from convexity. A plausible implication is that many partial Finsler examples should be studied first through their truncations when one wants access to positive-definite tools.

The geometric role of the indicatrix is unchanged but enriched by the slit. In a partial Minkowski space, the indicatrix

$\mathcal M\subset A$0

and solid indicatrix $\mathcal M\subset A$1 determine the geometry, and homogeneity implies that the indicatrix determines $\mathcal M\subset A$2 (Davis et al., 29 Aug 2025). For bipartite spaces, the paper introduces the bipartite indicatrix union

$\mathcal M\subset A$3

where $\mathcal M\subset A$4 is the fixed-point set under scaling (Davis et al., 29 Aug 2025). This enlarged indicatrix object reflects the fact that the slit can carry geometrically distinguished nonzero directions.

3. Anchored bundles, foliations, and induced leafwise Finsler geometry

Pelletier’s construction places partial Finsler geometry on a foliated anchored bundle rather than directly on $\mathcal M\subset A$5. The anchor $\mathcal M\subset A$6 determines the distribution $\mathcal M\subset A$7, and a foliated anchored bundle is characterized by the existence of an almost Lie bracket making $\mathcal M\subset A$8 into a pre-Lie algebroid; equivalently, the module $\mathcal M\subset A$9 is involutive and defines a Stefan–Sussmann foliation (Pelletier, 2014).

The central point is that the resulting geometry is leafwise. If $F:\mathcal M\to [0,+\infty[$0 is a leaf of the foliation defined by $F:\mathcal M\to [0,+\infty[$1, then the partial Finsler metric induces a genuine Finsler metric $F:\mathcal M\to [0,+\infty[$2 on the quotient bundle

$F:\mathcal M\to [0,+\infty[$3

(Pelletier, 2014). In the special case $F:\mathcal M\to [0,+\infty[$4, this is simply the restriction to $F:\mathcal M\to [0,+\infty[$5. The paper emphasizes that all the relevant constructions depend only on the foliation induced by the anchor and not on the particular almost Lie bracket used to realize a pre-Lie algebroid structure (Pelletier, 2014).

This leafwise invariance is implemented through the prolongation bundle. Given a fibered manifold, the prolongation is

$F:\mathcal M\to [0,+\infty[$6

with adapted local basis

$F:\mathcal M\to [0,+\infty[$7

When a pre-Lie algebroid bracket is chosen, it prolongs to a bracket $F:\mathcal M\to [0,+\infty[$8 satisfying

$F:\mathcal M\to [0,+\infty[$9

(Pelletier, 2014). A major structural statement is that on each leaf $F$0, the quotient

$F$1

identifies canonically with $F$2, so the prolonged geometry descends to the ordinary tangent geometry of the leaf (Pelletier, 2014).

This framework generalizes both classical Finsler geometry and Lie algebroid Finsler geometry. When $F$3 and $F$4, one recovers the standard spray, nonlinear connection, Chern connection, and flag curvature (Pelletier, 2014). The new content is that these constructions can be made globally on a conic domain of an anchored bundle while their geometric meaning remains leafwise canonical.

4. Connections, sprays, and Chern-type structures

A partial Finsler metric $F$5 determines a $F$6-homogeneous Lagrangian $F$7, hence a canonical semispray and nonlinear connection. For a semispray $F$8, the associated nonlinear connection is

$F$9

and in an adapted basis its coefficients take the Finsler-type form

$\mathcal M$0

(Pelletier, 2014). In the partial Finsler case this is the Finsler connection $\mathcal M$1, whose geodesics are the integral curves of the canonical semispray $\mathcal M$2 and satisfy

$\mathcal M$3

(Pelletier, 2014). The paper states that $\mathcal M$4 is the unique connection that is simultaneously $\mathcal M$5-metric and Lagrangian with respect to $\mathcal M$6 (Pelletier, 2014).

The corresponding Chern connection is given by a uniqueness theorem. For a partial Finsler pre-Lie algebroid $\mathcal M$7, there exists a unique linear connection

$\mathcal M$8

that is torsion-free,

$\mathcal M$9

and almost $F(\lambda u)=\lambda F(u)$0-compatible,

$F(\lambda u)=\lambda F(u)$1

(Pelletier, 2014). This is the partial-Finsler analogue of the classical Chern connection, and it restricts on each leaf to the classical Chern connection of the induced Finsler metric $F(\lambda u)=\lambda F(u)$2 (Pelletier, 2014).

A complex analogue exists on holomorphic Lie algebroids. There, a complex Finsler structure is a real-valued function on the slit algebroid $F(\lambda u)=\lambda F(u)$3, smooth on $F(\lambda u)=\lambda F(u)$4, satisfying

$F(\lambda u)=\lambda F(u)$5

with positive-definite Hermitian Hessian

$F(\lambda u)=\lambda F(u)$6

(Ionescu, 2017). The Chern-Finsler nonlinear connection is defined by

$F(\lambda u)=\lambda F(u)$7

and the associated linear coefficients are

$F(\lambda u)=\lambda F(u)$8

which Theorem 2.8 identifies as the coefficients of the Chern-Finsler connection on the holomorphic Lie algebroid (Ionescu, 2017). Although the paper does not explicitly use the phrase partial Finsler manifold in exactly the same way as the real slit-based literature, it constructs the same package on a slit algebroid domain and is explicitly described as extending complex Finsler geometry to the holomorphic Lie algebroid setting (Ionescu, 2017).

5. Low regularity, partial smoothness, and smoothing theory

Partiality in Finsler geometry is not limited to slit domains. It also appears through weakened regularity assumptions. A low-regularity Myers–Steenrod theorem shows that if a Finsler metric is $F(\lambda u)=\lambda F(u)$9 or more generally $S\subset TM$00-partially smooth, then distance-preserving bijections have the same regularity behavior as in the Riemannian case after passage to the Binet–Legendre metric (Matveev et al., 2016).

For a Finsler structure on a domain $S\subset TM$01, the Binet–Legendre metric is defined by the inverse matrix of

$S\subset TM$02

where $S\subset TM$03 (Matveev et al., 2016). The principal theorem states that if $S\subset TM$04 is a distance-preserving bijection between Finsler manifolds and the associated Binet–Legendre metrics are locally $S\subset TM$05, with $S\subset TM$06, then

$S\subset TM$07

(Matveev et al., 2016). Theorem B′ shows that if $S\subset TM$08 is $S\subset TM$09-partially smooth and satisfies

$S\subset TM$10

then the Binet–Legendre metric is of class $S\subset TM$11 (Matveev et al., 2016). Corollary C′ then upgrades any distance-preserving bijection between such partially smooth Finsler manifolds to a

$S\subset TM$12

(Matveev et al., 2016). This indicates that a robust differential geometry survives under fiberwise “twisting” regularity.

At the opposite end of the regularity scale, a $S\subset TM$13-Finsler structure is a continuous function

$S\subset TM$14

whose restriction to each tangent space is an asymmetric norm (Fukuoka et al., 2019). A two-stage mollifier smoothing, first in the fibers and then in the base, constructs smooth Finsler structures $S\subset TM$15 converging uniformly on compact subsets to $S\subset TM$16. The global smoothing is

$S\subset TM$17

and Theorem 6.3 states that for any $S\subset TM$18-Finsler structure, each $S\subset TM$19 is a Finsler structure and

$S\subset TM$20

(Fukuoka et al., 2019). When the original $S\subset TM$21 is already smooth, the Chern, Cartan, Hashiguchi, and Berwald connections of $S\subset TM$22 converge uniformly on compact subsets of $S\subset TM$23 to those of $S\subset TM$24, and the same holds for flag curvature (Fukuoka et al., 2019).

This suggests a broad methodological principle: partially smooth, continuous, or piecewise smooth Finsler-type structures can often be studied by passage to canonical Riemannian averages or by smooth approximation. The papers do not identify all such objects with partial Finsler manifolds in a single formal sense, but they supply compatible analytical mechanisms for enlarging the Finsler category.

6. Examples, rigidity phenomena, and relations to conic Finsler geometry

The main explicit examples of partial Finsler manifolds in the slit-based sense are bipartite spaces. Given a Riemannian manifold $S\subset TM$25 with

$S\subset TM$26

and a symmetric nonnegative tensor $S\subset TM$27 whose eigenvalues lie in $S\subset TM$28, define

$S\subset TM$29

If

$S\subset TM$30

then

$S\subset TM$31

are partial Finsler manifolds (Davis et al., 29 Aug 2025). These examples show concretely why the slit must be allowed to contain nonzero directions: $S\subset TM$32 may fail to be smooth precisely on the scaling-fixed locus.

Special cases include Randers, $S\subset TM$33, and $S\subset TM$34 spaces. Randers norms have the form

$S\subset TM$35

and are standard Finsler norms (Davis et al., 29 Aug 2025). The $S\subset TM$36 spaces are bipartite spaces with

$S\subset TM$37

while the $S\subset TM$38 spaces are given by

$S\subset TM$39

(Davis et al., 29 Aug 2025). The paper states that the $S\subset TM$40 spaces are partial Finsler spaces with slit along the orthogonal complement of $S\subset TM$41, and that $S\subset TM$42 is always almost Minkowski, while $S\subset TM$43 is almost Minkowski only in dimension $S\subset TM$44 (Davis et al., 29 Aug 2025). It also proves that the bipartite indicatrix union of the $S\subset TM$45 spaces is isomorphic to the union of the indicatrices of the Randers spaces (Davis et al., 29 Aug 2025).

A different source of partiality is conic Finsler geometry. A conic Finsler manifold is a Finsler structure defined only on an open conic subset $S\subset TM$46, closed under positive scaling (Youssef et al., 2018). The paper on semi-concurrent vector fields emphasizes that non-Riemannian examples often survive only in this conic setting. In particular, it gives conic examples on domains such as

$S\subset TM$47

or similar conic domains, and shows that semi-concurrent vector fields can occur there without forcing a Riemannian metric (Youssef et al., 2018). The broader conclusion stated in that paper is a conjecture that there is no regular Finsler non-Riemannian metric admitting a semi-concurrent vector field, whereas conic Finsler spaces do admit such non-Riemannian examples (Youssef et al., 2018).

Rigidity phenomena also occur in the regular case. For a $S\subset TM$48-dimensional Finsler manifold $S\subset TM$49, if there exists a nonconstant smooth function $S\subset TM$50 satisfying

$S\subset TM$51

then $S\subset TM$52 is Riemannian (Faghfouri et al., 2015). The proof reduces the condition to a PDE forcing $S\subset TM$53 to be quadratic in the fiber variables, hence forcing vanishing Cartan torsion (Faghfouri et al., 2015). The question remains open for $S\subset TM$54, though the authors state that they have not found a counterexample and conjecture the answer may still be yes (Faghfouri et al., 2015). The semi-concurrent framework later subsumes this condition as the weakest of several related contraction conditions and recovers the two-dimensional conclusion, while extending Riemannian rigidity to additional classes such as reversible $S\subset TM$55-dimensional, C-reducible, and certain Berwaldian manifolds (Youssef et al., 2018).

These results mark an important distinction. On regular Finsler domains, several contraction or concurrency conditions force collapse to the Riemannian case. On conic or otherwise partial domains, non-Riemannian behavior persists. This suggests that partiality is not merely a technical inconvenience but a genuine geometric mechanism by which anisotropy can survive.

7. Characteristic tensors, analytic structures, and broader significance

Recent work develops intrinsic tensors adapted specifically to partial and almost Finsler settings. For bipartite spaces, starting from

$S\subset TM$56

the paper derives tensorial identities involving the Cartan tensor and mean Cartan torsion and defines a characteristic tensor

$S\subset TM$57

proving that $S\subset TM$58 vanishes for bipartite spaces (Davis et al., 29 Aug 2025). In the Randers limit $S\subset TM$59, this reduces to the usual Matsumoto tensor

$S\subset TM$60

while in the Riemannian limit $S\subset TM$61 it reduces to the Cartan torsion (Davis et al., 29 Aug 2025). For $S\subset TM$62 spaces the simplified tensor

$S\subset TM$63

vanishes (Davis et al., 29 Aug 2025). These constructions provide classification tools specifically adapted to slit-based, potentially non-positive-definite Finsler geometry.

Analytically, Finsler Sobolev theory shows how standard PDE frameworks extend once one has a sufficiently regular Finsler structure. On a $S\subset TM$64 Finsler manifold, the Busemann volume form is

$S\subset TM$65

and the osculating Riemannian metric $S\subset TM$66 is defined by

$S\subset TM$67

(Bidabad et al., 2013). Sobolev spaces $S\subset TM$68 are then built using the Levi-Civita connection of $S\subset TM$69, and density theorems show that on forward geodesically complete, connected, reversible $S\subset TM$70 Finsler manifolds,

$S\subset TM$71

(Bidabad et al., 2013). On compact domains with $S\subset TM$72 boundary, $S\subset TM$73 is dense in $S\subset TM$74 for $S\subset TM$75 (Bidabad et al., 2013). These results do not use the term partial Finsler manifold, but they illustrate how analytical machinery can be transferred to Finsler contexts through canonical measure and osculating metric constructions.

Taken together, the cited works delineate several overlapping enlargements of the classical Finsler category. One enlargement is domain-theoretic: norms defined only off a nontrivial conic slit, possibly with indefinite fundamental tensor, as in partial and almost Finsler manifolds (Davis et al., 29 Aug 2025). A second is algebroid-based: partial Finsler metrics on conic subsets of anchored bundles, whose geometry is intrinsically leafwise (Pelletier, 2014, Ionescu, 2017). A third is regularity-based: partially smooth or merely continuous Finsler structures that can be controlled via the Binet–Legendre metric or mollifier smoothing (Matveev et al., 2016, Fukuoka et al., 2019). Across these settings, the recurring theme is that the standard Finsler apparatus—fundamental tensor, Cartan tensor, sprays, Chern-type connections, curvature, and rigidity criteria—remains meaningful after suitable modifications, but its scope is determined by the geometry of the allowed directions, the positivity of the Hessian, and the regularity class in which the norm is defined.