Study of Matsumoto maps on foliated bundles over hyperbolic manifolds.
problem Characterizing ergodic harmonic measures on foliated bundles.
method Analysis of actions of hyperbolic manifold groups on the circle.
result Suspension of actions with non-discrete images cannot admit Matsumoto maps of type I.
Counterexample disproves Matsumoto's conjecture on Finsler manifolds.
problem Matsumoto's conjecture on absolute vs. relative lengths in Finsler manifolds.
method Provided a counterexample to disprove the conjecture.
result Matsumoto's conjecture is false for certain Finsler manifolds.
In this paper, we study the second approximate Matsumoto metric on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.
The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally Minkowskian.
In this paper, we discuss the invariant properties of curvatures effected by the Matsumoto's C or L-process. We find equivalent conditions on curvatures by comparing the difference between the corresponding curvatures of closely related connections. As an application, Matsumoto's L-process on Randers manifold is …
Study on Matsumoto's manifolds and their properties using Heegaard Floer homology.
problem Characterizing and understanding properties of Matsumoto's manifolds.
method Use of Heegaard Floer homology to study properties of Matsumoto's manifolds.
result No contractible bound of Mn(T2,3,K2) exists if n<2τ(K2). Study geometrical properties of Finsler space hypersurface with h-Matsumoto change.
problem Geometrical properties of hypersurface in Finsler space with h-Matsumoto change.
method Analyzing the Cartan connection for the transformed space.
result Found geometrical properties of hypersurface in Finsler space with h-Matsumoto change.
We generalize Matsumoto metrics with a special π-form and explore their geometric properties.
problem Exploring the geometric properties of generalized Matsumoto metrics with a special π-form.
method Considering a Finsler manifold with a concurrent π-vector field, we introduce a change in the metric and analyze its geometric properties.
result The generalized φ-Matsumoto metric can never be projectively related to the original metric.
In this paper, the necessary and sufficient conditions for Matsumoto metrics F=α−βα2 to be Einstein are given. It is shown that if the length of β with respect to α is constant, then the Matsumoto metric F is an Einstein metric if and only if α is Ricci-flat and β is parallel with respect to α. …
Research disproves Matsumoto's length conjecture using indicatrix geometry.
problem Matsumoto's conjecture about relative and Finsler lengths.
method Analysis of indicatrix geometry to refute the conjecture.
result Matsumoto's inequality between relative and Finsler lengths is false.
This paper contributes to the study of the Matsumoto metric F=alpha^2/beta, where the alpha is a Riemannian metric and the beta is a one form. It is shown that such a Matsumoto metric F is of scalar flag curvature if and only if F is projectively flat.
The paper examines Matsumoto change and its Cartan connection equivalence.
problem Analyzing the Matsumoto change and its geometric implications.
method Derived fundamental tensors and conditions for Cartan connection equivalence.
result Conditions for Cartan connection coefficients to be the same for both spaces.
In this paper we consider invariant Matsumoto metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces then we give the flag curvature formula of them. Also we study the special cases of naturally reductive spaces and bi-invariant metrics. We end the article by giving …
The projective algebra p(M;F) (i.e the collection of all projective vector fields)of a Finsler space (M;F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket. The projective algebra of Einstein metrics has been perpetually studied from physical and geometrical approaches. Here, the projective alg…
The paper proves no exotic actions of diffeomorphism groups on 1-manifolds.
problem The study tackles the exotic actions of diffeomorphism groups on 1-dimensional manifolds.
method The approach involves showing any nontrivial homomorphism has a standard form, with countably many embeddings and conjugate actions.
result The groups Diff^r_c(M) have no countable index subgroups, solving a conjecture of Matsumoto.
Researchers find multiple supporting minimal Lefschetz pencils for a complex fibration.
problem Finding multiple supporting minimal Lefschetz pencils for a specific Lefschetz fibration.
method Analyzing maximal disjoint (−1)-sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration. result The Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils.
Constructs genus-3 Lefschetz fibrations and exotic 4-manifolds with specific b2+ values.
problem Creating exotic 4-manifolds with specific second Betti number.
method Explicit construction of genus-3 Lefschetz fibrations and use of Luttinger surgery and twisted fiber sum.
result Explicit construction of exotic minimal symplectic 4-manifolds with b2+=3. The paper defines conditions for projective changes between specific Finsler metrics.
problem Characterizing projective changes between two (α,β)-metrics. method Analyzing conditions for projective change between (α,β)-metrics and Matsumoto metrics. result Conditions for projective changes between (α,β)-metrics and Matsumoto metrics are identified. The study characterizes minimal surfaces in a specific three-dimensional metric space and finds that planes are the only minimal surfaces.
problem Characterizing minimal surfaces in a specific three-dimensional metric space.
method Obtained partial differential equations to characterize minimal surfaces and proved that planes are the only such surfaces.
result Planes are the only minimal surfaces in the Matsumoto space.
In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly.
In [2], the first author constructed the first known examples of exotic minimal symplectic $\CP#5\CPb$ and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to $3\CP#7\CPb$. The construction in [2] uses Y. Matsumoto's genus two Lefschetz fibrations on $M = \mathbb{T}^{2}\times \mathbb{S}^{2} #4\C…
Fintushel-Stern's knot surgery gave many pairs of exotic manifolds, which are homeomorphic but non-diffeomorphic. We show that if an elliptic fibration has two parallel, oppositely oriented vanishing circles (for example S2×S2 or Matsumoto's S4), then the knot surgery gives rise to standard manifolds. The …
In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consi…
The paper introduces a new linking form for 3-manifolds in S4.
problem Embedding rational homology 3-spheres in S4. method Triple torsion linking form, higher order intersection form on 4-manifolds.
result An embedding obstruction for rational homology spheres in S4. In this paper, we study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metric…
Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
The paper constructs genus two Lefschetz fibrations with specific topological properties.
problem Constructing Lefschetz fibrations with specific topological invariants.
method Applying lantern substitutions and fiber sums to Matsumoto's genus two Lefschetz fibrations.
result The fibrations have symplectic minimal total spaces and symplectic Kodaira dimension 2.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
The study introduces new tensors for almost Finsler manifolds and analyzes their properties.
problem Defining and analyzing new types of Finsler manifolds.
method Introducing new Finsler manifolds, studying their properties, and deriving characteristic tensors.
result Characteristic tensors for almost Finsler manifolds have been generalized and their properties have been studied.
New Finsler metrics derived from pedal curves.
problem Constructing new Finsler metrics.
method Using pedal curves or surfaces of other curves/surfaces.
result Generalization of slope metric.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
Study harmonic measures and rigidity in Seifert 3-manifolds using S1-connections.
problem Rigidity of foliations on Seifert 3-manifolds with maximal Euler number.
method Using S1-connections and harmonic measures, proving the Gauss--Bonnet formula and rigidity results. result A harmonic measure on the suspension bundle of the action with maximal Euler number has rigidity, closely related to the Poisson kernel.
Study on flag curvatures of specific metrics on Lie groups.
problem Investigate flag curvatures of left invariant (α,β)-metrics and Randers metrics. method Explicit formulas and necessary/sufficient conditions for Berwald and Douglas types.
result Flag curvatures of (α,β)-metrics and Randers metrics on Lie groups can be zero, positive, or negative. In the paper `Automorphic functions for a Whitehead-complement group', [Osaka J Math 43 (2006) 63-77] Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3-dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automo…
The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
Constructs Einstein ACH metrics with CR structures, proving CR GJMS operators exist.
problem Constructing Einstein ACH metrics with specified CR structures.
method Refined Matsumoto's construction, solving Einstein equation to infinite order.
result Self-dual Einstein ACH metrics constructed for CR structures.
The concept of h-vector was introduced by H. Izumi in 1980. Recently we have obtained the Cartan connection for the Finsler space whose metric is given by Kropina change with an h-vector. In 1985, M. Matsumoto studied the theory of Finsler hypersurface. In this paper, we derive certain geometrical properties of a Finsl…
This work analyzes minimum-time navigation on Riemannian manifolds using Finsler geometry.
problem Minimum-time navigation on Riemannian manifolds.
method Finsler geometry, introducing superwind concept, extending (α,β)-metrics. result Time-minimizing geodesics and necessary/sufficient conditions for strong convexity.
On a Finsler manifold (M,L), we consider the change L⟶Lˉ(x,y)=eσ(x)L(x,y)+β(x,y), which we call a β-conformal change. This change generalizes various types of changes in Finsler geometry: conformal, C-conformal, h-conformal, Randers and generalized Randers changes. Under this change, we …
Characterizes Fuchsian actions via topological rigidity.
problem Understanding Fuchsian actions on the circle.
method Proves rigidity of certain representations of surface groups.
result Characterizes Fuchsian actions by topological rigidity.
Study shows non-locally-flat concordance elements in knot theory.
problem Understanding non-locally-flat concordance elements in knot theory.
method Utilized Heegaard Floer homology.
result Cokernel of the map from smooth knot concordance group to homology concordance group is infinitely generated and contains elements of infinite order.
The problem of pursuing a moving target is always one of the main topics in navigation. In the literatures, there are two well-known algorithms called Pure Pursuit and Pure Rendezvous navigation in the 3-dimensional space R3. In this paper, these two methods are combined to introduce a novel family of pursu…
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…
Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …
Computes the component group of real reductive groups.
problem Computing the component group of real reductive groups.
method Using structure results for real loci of algebraic groups and Galois cohomology.
result Explicit elements representing all connected components of G(R). A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
Study on Finsler metrics finds no P-reducible metrics with vanishing S-curvature.
problem Existence of P-reducible metrics with vanishing S-curvature.
method Investigation of generalized P-reducible metrics and proving their reduction to Berwald or C-reducible metrics.
result No concrete P-reducible (α,β)-metric with vanishing S-curvature exists. The paper explores generalized Douglas-Weyl metrics and their properties.
problem Characterizing and understanding generalized Douglas-Weyl metrics.
method Analyzing properties of (α,β)-metrics and proving conditions for being generalized Douglas-Weyl metrics. result Generalized Douglas-Weyl metrics are Berwald metrics under certain conditions.