Modern geometry--methods and applications
Modern geometry--methods and applications
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DOI：
10.1007/978-1-4684-9946-9
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
B. Dubrovin;S. Novikov;A. Fomenko
中科院分区：
文献类型：
--
作者：
B. Dubrovin;S. Novikov;A. Fomenko
直到最近，黎曼几何和基本拓扑学还没有被包括在内，甚至数学系也没有将其作为大学数学教育的必修科目。标准课程的经典微分几何的曲线和曲面，而不是给予（仍然是在某些地方）已逐渐被视为时代错误。然而，迄今为止还没有一致的协议，究竟如何这些课程应带来了最新的，也就是说，哪些部分的现代几何应被视为绝对必要的现代数学教育，什么可能是适当的水平抽象的阐述。1971年，在莫斯科州立大学力学和数学系力学系开始设计现代化的几何课程。其论述的主题和抽象程度是由这样一种观点决定的，即除了曲线和曲面的几何之外，下列主题在数学应用的各个领域中肯定是有用的（特别是在弹性和相对论，仅举两例），因此是必不可少的：张量理论（包括它们的协变微分）;黎曼曲率;测地线和变分法（包括守恒定律和哈密顿形式主义）;反对称张量的特殊情况（即
Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (ie