Modern Geometry_ Methods And Applications: Part Ii: The Geometry And Topology Of Manifolds (graduate Texts In Mathematics)
by A.T. Fomenko /
2012 / English / PDF
20.8 MB Download
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 (i. e.
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 (i. e.
