Legendre transformation and lifting of multi-vectors

一类广义 Frobenius 流形的 Legendre 变换及相应的可积层级

Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchies

本文研究一类广义 Frobenius 流形在 Legendre 型变换下的可积层级关系，证明 Legendre 扩展主层级及其半单条件下的拓扑形变均由同一线性互反变换联系。

A slow and careful Legendre transformation for singular Lagrangians

A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate characterizations of all structures are provided. Intrinsic constructions of most of the objects are given. Examples of applications to a number of physical systems is given

V-representation for normality equations in geometry of generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by classical and/or generalized Legendre transformation. However, in this case they appear to be written in p-representation, i. e. in terms of momentum covector and its components. The goal of present paper is to transform normality equations back to v-representation, which is more nat…

Frobenius流形的Legendre型变换的解析理论

Analytic theory of Legendre-type transformations for a Frobenius manifold

本文证明了Frobenius流形在Dubrovin的Legendre型变换下保持单值性数据、Stokes矩阵和中心联络矩阵不变，并给出了该不变性在单值性数据获取和配分函数识别中的应用。

Homogenous Lagrangian systems

The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativistic mechanics. The dynamics of relativistic systems must be formulated in terms of implicit differential equations in the phase space and not in terms of Hamiltonian vector fields. The constrained Hamiltonian systems introduced by Dirac [1] are not general enough to cover some i…

The Ribaucour transformation in Lie sphere geometry

We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer--Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms.

关于绷紧性在李球面变换下不变性的札记

Notes on the Invariance of Tautness Under Lie Sphere Transformations

本文阐述绷紧性在李球面变换下的不变性证明，通过将球面子流形的绷紧性推广为 Legendre 子流形的李绷紧性，并证明二者等价及变换不变性。

Universal lifting theorem and quasi-Poisson groupoids

We prove the universal lifting theorem: for an $α$-simply connected and $α$-connected Lie groupoid $\gm$ with Lie algebroid $A$, the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $\gm$. As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair $(D, G)$ associated to a Manin quasi-triple $(\mathf…

The Binet-Legendre Metric in Finsler Geometry

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation of the Finsler metric $F$. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a q…

Mirror symmetry without corrections

We give geometric explanations and proofs of various mirror symmetry conjectures for $T^{n}$-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, $H^{p,q}$'s, Yukawa couplings; (ii) sl(2)xsl(2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) A- and B-connections, supersymmetric A- and B-cycles, correlation functions. We also study (ii) for $T^{n}$-invariant hy…