Geometry and Analysis on Nonholonomic structures on manifolds

[摘要]获得：DMS 1406193，首席研究员：Igor zelenkoi非完整结构，如支架生成分布和亚黎曼结构，在控制理论中自然出现，并应用于机器人（类汽车机器人），力学（球轴承结构）和视觉感知模型。分布也是微分方程几何理论和复空间实子流形几何（柯西-黎曼几何）中的基本对象。了解几何结构的不变量直至环境流形的微分同态，或者非正式地说，了解与这种结构的局部坐标表示无关的量，通常可以提供有关几何结构各种定性性质的基本信息。在过去的十年中，我们发展了一种新的变分方法来统一构造一类非常广泛的几何结构的微分不变量。这种方法起源于最优控制理论和辛几何。与所有先前存在的方法（包括经典的Cartan等价方法和Tanaka理论）相比，它显著地扩展了正则框架和微分不变量可以显式和一致构造的非完整结构集。该项目的目的是使用这些不变量来解决一些以前没有这样普遍考虑过的自然问题。该项目的研究结果将为控制理论和数学物理提供新的几何工具，为计算状态反馈和规范不变量提供有效和统一的算法，并为相应变分问题的极值提供显式几何最优性条件。我们将开发一个特殊的基于MAPLE的软件包来计算我们的不变量，并将理论应用于实际兴趣的控制和机械系统的定性研究。上述变分方法的主要新观点是，流形上非完整结构的几何研究可以简化为旗变中曲线的更简单的（外在的）几何。根据这些曲线，我们得到了原始结构的一个新的离散基本不变量，称为标志符号，我们有一个明确的算法来构建原始结构的规范框架，它只依赖于首先固定这个离散信息。然而，这些框架及其产生的不变量的主要性质还远未被理解。该项目将解决的问题包括：(1)明确描述具有给定标志符号的最对称模型的对称组；(2)探索得到的正则框架何时为Cartan连接；(3)在这些框架产生的所有不变量中区分基本不变量集。为了达到这些目标，我们将使用表示理论、代数几何和李代数上同调理论中的各种工具和技术。项目的另一个主题是亚黎曼几何中的比较定理。与黎曼情况相比，这里有几个本质上的差异，这对获得经典劳赫和邦纳-迈尔斯比较定理的尖锐类比构成了严重的障碍。为了克服这些障碍，我们建议研究比较定理的无穷小版本，即给定一个子黎曼度量来描述在这个度量与所有子黎曼度量空间的切空间中的方向，其中沿相应极值的连续共轭点变得更近。我们还将研究亚黎曼度量空间上的自然流，在这个空间上无穷小比较定理成立。

AbstractAward: DMS 1406193, Principal Investigator: Igor ZelenkoStructures of nonholonomic nature such as bracket generating distributions and sub-Riemannian structures appear naturally in Control Theory with applications to Robotics (car-like robots), Mechanics (ball bearing structures) and the models of visual perception. Distributions are also basic objects in the geometric theory of differential equations and in geometry of real submanifolds of complex spaces (Cauchy-Riemann geometry). The knowledge of invariants of a geometric structure up to diffeomorphisms of the ambient manifolds or, informally speaking, of the quantities which are independent of local coordinate representations of such structures often gives the essential information about various qualitative properties of the geometric structures. In the last decade we developed the novel variational approach for the unified construction of the differential invariants for a very wide class of geometric structures. This approach takes its origin in Optimal Control Theory and Symplectic Geometry. It extends significantly the set of the nonholonomic structures for which the canonical frames and differential invariants can be constructed explicitly and uniformly compared with all previously existing approaches (including the classical Cartan method of equivalence and the Tanaka theory from 1970). The aim of the project is to use these invariants in order to solve a number of natural problems that were not considered in such generality before. The results of this project will bring new geometric tools for Control Theory and Mathematical Physics by providing efficient and uniform algorithms for computing state-feedback and gauge invariants and explicit geometric optimality conditions for extremals of the corresponding variational problems. We will develop a special MAPLE based software package for computation of our invariants and apply the theory to the qualitative study of control and mechanical systems of practical interest.The main new point of the aforementioned variational approach is that the study of geometry of nonholonomic structures on manifold can be reduced to a simpler (extrinsic) geometry of curves in flag varieties. In terms of these curves we obtain a new discrete basic invariant of the original structure, called the flag symbol and we have an explicit algorithm for construction of the canonical frame for our original structure that depends only on first fixing this discrete information. However, the main properties of these frames and of the invariants they produce are far of being understood. Among the problems that will be addressed in the project are (1) explicit description of the group of symmetries of the most symmetric models with given flag symbol; (2) exploring when the obtained canonical frames are Cartan connections; (3) distinguishing the fundamental set of invariants among all invariants produced by these frames. To reach these goals we will use various tools and techniques from Representation Theory, Algebraic Geometry, and cohomological theory of Lie algebras. Another main theme of the project is comparison theorems in sub-Riemannian geometry. Several essential differences here compared with the Riemannian case form serious obstacles in obtaining sharp analogs of the classical Rauch and Bonnet-Myers comparison theorems. To overcome these obstacles we propose to work on the infinitesimal version of comparison theorems, i.e. given a sub-Riemannian metric to describe directions in the tangent space at this metric to the space of all sub-Riemannian metrics in which consecutive conjugate points along the corresponding extremals become closer. We also shall study natural flows on the space of sub-Riemannian metrics along which the infinitesimal comparison theorem holds.