A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.

We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

In this paper we outline the foundations of Homological Mirror Symmetry for manifolds of general type. Both Physics and Categorical prospectives are considered.

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.

Une homotopie régulière ${\varphi }_t:\Delta \to (M,\omega )$, $t\in [0,1]$, dans une variété symplectique est dite inactive si en chaque point le déplacement infinitésimal est $\omega$-orthogonal à l’espace tangent de l’objet déplacé. Si $\Delta$ est un polyèdre de $M^{2n}$ de dimension $\<n$ et si $U$ est un ouvert de $M$, toute homotopie de $\Delta \hookrightarrow M$ jusqu’à $\Delta \to U$ est déformable en une homotopie régulière inactive. On donne une application à l’engouffrement en géométrie symplectique.

Summary: All algebraic objects in this note will be considered over a fixed field $k$ of characteristic zero. If not stated otherwise, all operads live in the category of differential graded vector spaces over $k$. For standard terminology concerning operads, algebras over operads, etc., see either the original paper by J. P. May [“The geometry of iterated loop spaces”, Lect. Notes Math. 271 (1972; Zbl 0244.55009)], or an overview [J.-L. Loday, “La renaissance des opérads”, Sémin. Bourbaki 1994/95,...

The paper is concerned with homotopy concepts in the category of chain complexes. It is part of the author’s program to translate [J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347, Springer-Verlag (1973; Zbl 0285.55012)] from topology to algebra.In topology the notion of operad extracts the essential algebraic information contained in the following example (endomorphism operad).The endomorphism operad ${ℰ}_X$ of a based space $X$ consists...

Let $F({ℝ}^n,k)$ denote the configuration space of pairwise-disjoint $k$-tuples of points in ${ℝ}^n$. In this short note the author describes a cellular structure for $F({ℝ}^n,k)$ when $n\ge 3$. From results in [F. R. Cohen, T. J. Lada and J. P. May, The homology of iterated loop spaces, Lect. Notes Math. 533 (1976; Zbl 0334.55009)], the integral (co)homology of $F({ℝ}^n,k)$ is well-understood. This allows an identification of the location of the cells of $F({ℝ}^n,k)$ in a minimal cell decomposition. Somewhat more detail is provided by the main result here,...

Regarding the generalized Tanaka-Webster connection, we considered a new notion of $${𝔇}^{\perp }$$ -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $${𝔇}^{\perp }$$ -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2\left({ℂ}^{m+2}\right)$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ and prove non-existence of real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ with generalized Tanaka-Webster...