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The Foundations Of Differential Geometry

This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

Integral formulas in Riemannian geometry

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

The theory of Lie derivatives and its applications

The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

Differential Geometry of Special Mappings

One of the present authors defined the infinitesimal harmonictransformation in a Riemannian manifold. This paperis devoted to the study of the local and global geometryinfinitesimal harmonic transformations.

https://link.springer.com/content/pdf/10.1023/A:1024753028255.pdf

Geometry of Infinitesimal Harmonic Transformations

where aα = aαβg , and the gij are the elements of the inverse matrix to gij , and the commas denote covariant differentiation with respect to the connection on (M,g). These equations are linear, so that their solutions form a linear vector space; the dimension of this space is called the degree of geodesic mobility of the metric g and denoted by p. The number p is a numerical characteristic of the cardinality of the geodesic class of the given metric gij (see [1]). It has long been known that metrics with large degree of mobility admit concircular covector fields [2]. Recall that a covector field φi on (M,g) is said to be concircular if φi,j = ρgij (2)

http://users.minet.uni-jena.de/~matveev/Datei/schandra.pdf

On the degree of geodesic mobility for Riemannian metrics

We prove several Liouville-type nonexistence theorems for higher-order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because, its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.

On higher-order Codazzi tensors on complete Riemannian manifolds

In spite of the abundance of publications on harmonic mappings of man- ifolds, at present there exists neither a theory of harmonic diffeomorphisms, nor a definition of infinitesimal harmonic transformation of a Riemannian manifold, to say nothing of the theory of groups of such transformations. In the paper, this gap is partially filled, and a new subject of investigations is announced.

/pdf/harmonic-diffeomorphisms-of-manifolds-2bwft0cfrl.pdf

I. G. Shandra

Papers

Differential Geometry of Special Mappings

Geometry of Infinitesimal Harmonic Transformations

On the degree of geodesic mobility for Riemannian metrics

On higher-order Codazzi tensors on complete Riemannian manifolds

Harmonic diffeomorphisms of manifolds