--- title: Partial Finsler Manifolds url: https://www.emergentmind.com/topics/partial-finsler-manifolds type: topic --- # Partial Finsler Manifolds 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 [1412.6742]. 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 [2508.21744]. 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 [1605.03850], [1802.02405], [1910.14331]. ## 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 \(\lambda>0\), and has positive-definite fundamental quadratic form \[ g_u(v,w):=\frac12 \frac{\partial^2}{\partial s\partial t}\Big(F^2(u+sv+tw)\Big)\Big|_{s=t=0} \] for every \(u\in \mathcal M_x\) [1412.6742]. 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 \(S\subset TM\). A partial Finsler manifold is a triple \[ (M,S,F) \] with \(S\) a closed conelike subset containing the zero section and \[ F:TM-S\to (0,\infty) \] smooth and homogeneous of degree \(1\) in each tangent space, such that each fiber restriction is a partial Minkowski norm [2508.21744]. In this framework, an almost Finsler manifold is a partial Finsler manifold whose fiberwise Hessian of \(F^2/2\) is positive definite on \(TM-S\), and a standard Finsler manifold is recovered by taking the slit to be exactly the zero section [2508.21744]. The corresponding local model is a partial Minkowski space \((V,S,F)\), where \(V\) is a finite-dimensional real vector space, \(S\subset V\) is closed and conelike, and \(F:V-S\to (0,\infty)\) is positively homogeneous: \[ F(\lambda y)=\lambda F(y),\qquad \lambda>0,\ y\in V-S. \] If the bilinear form \[ g_y(\cdot,\cdot) :=\frac12 \left.\frac{\partial^2}{\partial s\,\partial t}\,F^2(y+s u+t v)\right|_{s=t=0} \] is positive definite for every \(y\in V-S\), one obtains an almost Minkowski space [2508.21744]. 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 \((k,\alpha)\)-partial smoothness. A continuous Finsler metric \(F\) on \(U\subset \mathbb R^n\) is \((k,\alpha)\)-partially smooth if there exists a positively homogeneous, fiberwise bilipschitz map \[ A: U\times \mathbb R^n \to \mathbb R^n \] such that, for all \(u\in S^{n-1}\), \[ \|A_u\|_{C^{k,\alpha}(U)}+\|J_u\|_{C^{k,\alpha}(U)}+\|h_u\|_{C^{k,\alpha}(U)}\le C, \] where \[ A_u(x)=A(x,u),\qquad J_u(x)=|\det D A_x(u)|,\qquad h_u(x)=F(x,A(x,u)). \] This allows a metric to become regular only after a fiberwise change of variables [1605.03850]. 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 \(TM-S\). With fiber coordinates \(y^j\), the fundamental tensor is \[ g_{jk}=\frac12 (F^2)_{jk}, \] the Cartan tensor is \[ C_{jkl}=\frac14 (F^2)_{jkl}, \] the Hilbert form is \[ p_j = F_j, \] the angular metric is \[ h_{jk}=F\,F_{jk}, \] and the mean Cartan torsion is \[ I_j = g^{kl}C_{jkl} \] [2508.21744]. Homogeneity yields the usual Euler identities, including \[ y^jF_j = F,\qquad y^jF_{jk}=0 \] [2508.21744]. In the anchored-bundle setting, the fundamental tensor is the vertical Riemannian metric \(g_F\), and the Cartan tensor is the symmetric trilinear form \[ A_u(v_1,v_2,v_3) =\frac12 \frac{\partial^3}{\partial s_1\partial s_2\partial s_3} F^2\!\left(u+\sum_{i=1}^3 s_i v_i\right)\Big|_{s_1=s_2=s_3=0}, \] satisfying \(A_u(u,v_2,v_3)=0\) [1412.6742]. 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 \((M,S,F)\), the slit tangent bundle is \(TM-S\), and the paper introducing almost and partial Finsler manifolds defines the extended slit \[ ES:=S\cup\{y\in TM-S\mid g_y\ \text{is not positive definite}\}. \] The triple \((M,ES,F)\) is then called the truncation and is always an almost Finsler manifold [2508.21744]. 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 \[ I=F^{-1}\{1\} \] and solid indicatrix \(F^{-1}[0,1]\) determine the geometry, and homogeneity implies that the indicatrix determines \(F\) [2508.21744]. For bipartite spaces, the paper introduces the bipartite indicatrix union \[ \widehat I = I^+\cup I^-\cup S^f, \] where \(S^f\) is the fixed-point set under scaling [2508.21744]. 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 \(TM\). The anchor \(p:A\to TM\) determines the distribution \(p(A)\subset TM\), and a foliated anchored bundle is characterized by the existence of an almost Lie bracket making \((A,M,p)\) into a pre-Lie algebroid; equivalently, the module \(p(\Gamma(A))\) is involutive and defines a Stefan–Sussmann foliation [1412.6742]. The central point is that the resulting geometry is leafwise. If \(N\) is a leaf of the foliation defined by \(p(A)\), then the partial Finsler metric induces a genuine Finsler metric \(F_N\) on the quotient bundle \[ A_N/K_N \;\xrightarrow{\;\cong\;}\; TN, \qquad K_N:=A_N\cap \ker p \] [1412.6742]. In the special case \(A\subset TM\), this is simply the restriction to \(TN\). 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 [1412.6742]. This leafwise invariance is implemented through the prolongation bundle. Given a fibered manifold, the prolongation is \[ \mathcal T M =\{(m,b,v)\in M\times A\times TM \mid p(b)=T_m\tau(v)\}, \] with adapted local basis \[ X_\alpha=(e_\alpha\circ \tau,\, p_\alpha^i \partial_{x^i}),\qquad V_\alpha=(0,\partial_{y^\alpha}). \] When a pre-Lie algebroid bracket is chosen, it prolongs to a bracket \([\ ,\ ]_{\mathcal T}\) satisfying \[ [X_\alpha,X_\beta]_{\mathcal T}=C_{\alpha\beta}^{\gamma}X_\gamma,\qquad [X_\alpha,V_\beta]_{\mathcal T}=0,\qquad [V_\alpha,V_\beta]_{\mathcal T}=0 \] [1412.6742]. A major structural statement is that on each leaf \(N\), the quotient \[ \mathcal T M_N / \big(K_{M_N}+J(K_{M_N})\big) \] identifies canonically with \(T(TN)\), so the prolonged geometry descends to the ordinary tangent geometry of the leaf [1412.6742]. This framework generalizes both classical Finsler geometry and Lie algebroid Finsler geometry. When \(A=TM\) and \(\mathcal M=TM\setminus\{0\}\), one recovers the standard spray, nonlinear connection, Chern connection, and flag curvature [1412.6742]. 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 \(F\) determines a \(2\)-homogeneous Lagrangian \(L=F^2\), hence a canonical semispray and nonlinear connection. For a semispray \(S\), the associated nonlinear connection is \[ N_S = -\mathcal L_S J, \] and in an adapted basis its coefficients take the Finsler-type form \[ N_\alpha^{\ \beta} = -\frac12\left(\frac{\partial S^\beta}{\partial y^\alpha} + C_{\alpha\gamma}^{\beta}y^\gamma\right) \] [1412.6742]. In the partial Finsler case this is the Finsler connection \(N_F\), whose geodesics are the integral curves of the canonical semispray \(S_F\) and satisfy \[ \dot x^i = p_\alpha^i\, y^\alpha,\qquad \dot y^\alpha = S^\alpha(x,y) \] [1412.6742]. The paper states that \(N_F\) is the unique connection that is simultaneously \(g_F\)-metric and Lagrangian with respect to \(F^2\) [1412.6742]. The corresponding Chern connection is given by a uniqueness theorem. For a partial Finsler pre-Lie algebroid \((A,M,p,[\ ,\ ]_A,F)\), there exists a unique linear connection \[ V:\Gamma(TM)\times \Gamma(A)\to \Gamma(A) \] that is torsion-free, \[ V_X \hat Y - V_Y \hat X = \widehat{[X,Y]_{\mathcal T}}, \] and almost \(g_F\)-compatible, \[ p(X)\big(g_F(\hat Y,\hat Z)\big) = g_F(V_X\hat Y,\hat Z)+g_F(\hat Y,V_X\hat Z) + A_F\big(\theta(X),\hat Y,\hat Z\big) \] [1412.6742]. 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 \(F_N\) [1412.6742]. A complex analogue exists on holomorphic Lie algebroids. There, a complex Finsler structure is a real-valued function on the slit algebroid \(E^\circ=E\setminus\{0\}\), smooth on \(E^\circ\), satisfying \[ F(z,\lambda u)=|\lambda|^2F(z,u),\qquad \lambda\in\mathbb C, \] with positive-definite Hermitian Hessian \[ h_{a\bar b}=\frac{\partial^2 F}{\partial u^a\,\partial \bar u^b} \] [1705.09095]. The Chern-Finsler nonlinear connection is defined by \[ N_a^{\ b}=h^{\bar c b}\, \delta_a\left(\frac{\partial F}{\partial \bar u^c}\right), \] and the associated linear coefficients are \[ L_{ab}^{\ c}=h^{\bar d c}\,\delta_a h_{b\bar d}, \] which Theorem 2.8 identifies as the coefficients of the Chern-Finsler connection on the holomorphic Lie algebroid [1705.09095]. 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 [1705.09095]. ## 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 \(C^{k,\alpha}\) or more generally \((k,\alpha)\)-partially smooth, then distance-preserving bijections have the same regularity behavior as in the Riemannian case after passage to the Binet–Legendre metric [1605.03850]. For a Finsler structure on a domain \(U\subset\mathbb R^n\), the Binet–Legendre metric is defined by the inverse matrix of \[ g^{ij}(x) = \frac{n+2}{\operatorname{Vol}(\Omega_x)} \int_{\Omega_x} v_i v_j\, dv, \] where \(\Omega_x=\{v\in \mathbb{R}^n : F(x,v)<1\}\) [1605.03850]. The principal theorem states that if \(\varphi : M_1 \to M_2\) is a distance-preserving bijection between Finsler manifolds and the associated Binet–Legendre metrics are locally \(C^{k,\alpha}\), with \(k+\alpha>0\), then \[ \varphi\in C^{k+1,\alpha}(M_1,M_2), \qquad\text{and}\qquad \varphi^*(F_2)=F_1 \] [1605.03850]. Theorem B′ shows that if \(F\) is \((k,\alpha)\)-partially smooth and satisfies \[ C^{-1}|v|\le F(x,v)\le C|v|, \] then the Binet–Legendre metric is of class \(C^{k,\alpha}\) [1605.03850]. Corollary C′ then upgrades any distance-preserving bijection between such partially smooth Finsler manifolds to a \[ C^{k+1,\alpha}\text{-diffeomorphism} \] [1605.03850]. This indicates that a robust differential geometry survives under fiberwise “twisting” regularity. At the opposite end of the regularity scale, a \(C^0\)-Finsler structure is a continuous function \[ F: TM \to [0,\infty) \] whose restriction to each tangent space is an asymmetric norm [1910.14331]. A two-stage mollifier smoothing, first in the fibers and then in the base, constructs smooth Finsler structures \(F_\varepsilon\) converging uniformly on compact subsets to \(F\). The global smoothing is \[ F_\varepsilon(x,y)=\sum_{\lambda}\psi_\lambda(x)\,F_{\varepsilon,\lambda}(x,y), \] and Theorem 6.3 states that for any \(C^0\)-Finsler structure, each \(F_\varepsilon\) is a Finsler structure and \[ F_\varepsilon\to F \quad \text{uniformly on compact subsets of } TM \] [1910.14331]. When the original \(F\) is already smooth, the Chern, Cartan, Hashiguchi, and Berwald connections of \(F_\varepsilon\) converge uniformly on compact subsets of \(TM\setminus 0\) to those of \(F\), and the same holds for flag curvature [1910.14331]. 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 \((M,\rho)\) with \[ \rho(y)=\sqrt{r_{jk}y^jy^k}, \] and a symmetric nonnegative tensor \(s_{jk}\) whose eigenvalues lie in \([0,1)\), define \[ \sigma(y)=\sqrt{s_{jk}y^jy^k}, \qquad F^\pm=\rho\pm \sigma. \] If \[ S=\bigcup_{x\in M} S_x,\qquad S_x=\ker(s_x), \] then \[ (M,S,F^+)\quad\text{and}\quad (M,S,F^-) \] are partial Finsler manifolds [2508.21744]. These examples show concretely why the slit must be allowed to contain nonzero directions: \(F^2\) may fail to be smooth precisely on the scaling-fixed locus. Special cases include Randers, \(\mathbf a\), and \(\mathbf b\) spaces. Randers norms have the form \[ F_\pm(y)=\rho(y)\pm \langle b,y\rangle_\rho \] and are standard Finsler norms [2508.21744]. The \(\mathbf a\) spaces are bipartite spaces with \[ F^\pm = \rho \pm \sigma,\qquad \sigma = \rho(b)\,\rho_\parallel, \] while the \(\mathbf b\) spaces are given by \[ F^\pm = \rho \pm \sigma,\qquad \sigma = \rho(b)\,\rho_\perp \] [2508.21744]. The paper states that the \(\mathbf a\) spaces are partial Finsler spaces with slit along the orthogonal complement of \(b\), and that \(F^+_{\mathbf b}\) is always almost Minkowski, while \(F^-_{\mathbf b}\) is almost Minkowski only in dimension \(2\) [2508.21744]. It also proves that the bipartite indicatrix union of the \(\mathbf a\) spaces is isomorphic to the union of the indicatrices of the Randers spaces [2508.21744]. A different source of partiality is conic Finsler geometry. A conic Finsler manifold is a Finsler structure defined only on an open conic subset \(A\subset TM\), closed under positive scaling [1802.02405]. 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 \[ D=TM-\{(x_i,y_i)\in TM \mid y_1^2+y_2^2\neq 0\} \] or similar conic domains, and shows that semi-concurrent vector fields can occur there without forcing a Riemannian metric [1802.02405]. 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 [1802.02405]. Rigidity phenomena also occur in the regular case. For a \(2\)-dimensional Finsler manifold \((M,F)\), if there exists a nonconstant smooth function \(f\) satisfying \[ \frac{\partial g^{ij}}{\partial y^k}\,\frac{\partial f}{\partial x^i}=0, \] then \(M\) is Riemannian [1509.09172]. The proof reduces the condition to a PDE forcing \(F^2\) to be quadratic in the fiber variables, hence forcing vanishing Cartan torsion [1509.09172]. The question remains open for \(n>2\), though the authors state that they have not found a counterexample and conjecture the answer may still be yes [1509.09172]. 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 \(3\)-dimensional, C-reducible, and certain Berwaldian manifolds [1802.02405]. 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 \[ F-\Delta=\rho,\qquad \Delta=\pm\sigma, \] the paper derives tensorial identities involving the Cartan tensor and mean Cartan torsion and defines a characteristic tensor \[ S_{jkl} = C_{jkl} -\frac{1}{\kappa} \sum_{(jkl)} \left(I_j+\frac{F^2}{(F-\Delta)\Delta}\,g^{kl}\Delta_k\Delta_{lj}\right) \left(h_{kl}-\frac{F^2}{\Delta}\Delta_{kl}\right), \] proving that \(S_{jkl}\) vanishes for bipartite spaces [2508.21744]. In the Randers limit \(\Delta_{jk}=0\), this reduces to the usual Matsumoto tensor \[ M_{jkl} = C_{jkl} -\frac{1}{n+1}\sum_{(jkl)} I_j h_{kl}, \] while in the Riemannian limit \(\Delta=0\) it reduces to the Cartan torsion [2508.21744]. For \(\mathbf b\) spaces the simplified tensor \[ B_{jkl} = C_{jkl} -\frac{1}{\kappa_b}\sum_{(jkl)} I_j\left(h_{kl}-\frac{F^2}{\Delta}\Delta_{kl}\right) \] vanishes [2508.21744]. 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 \(C^\infty\) Finsler manifold, the Busemann volume form is \[ d\nu_F = \sigma_F(x)\, dx^1\wedge \cdots \wedge dx^n, \] and the osculating Riemannian metric \(K\) is defined by \[ K_{ij}(x)= (n+2)\int_{I_x} y^i y^j\, d\nu_x \] [1310.8027]. Sobolev spaces \(H_k^p(M)\) are then built using the Levi-Civita connection of \(K\), and density theorems show that on forward geodesically complete, connected, reversible \(C^\infty\) Finsler manifolds, \[ C_c^\infty(M)\ \text{is dense in } H_1^p(M) \] [1310.8027]. On compact domains with \(C^r\) boundary, \(C^r(W)\cap C^0(\overline W)\) is dense in \(H_k^p(W)\) for \(k