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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

Exact Solutions of Einstein's Field Equations: Notation

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

Conformal Killing $L^{2}-$forms on complete Riemannian manifolds with nonpositive curvature operator

On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, …, n − 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M,g) is not zero.

Theorems of existence and of vanishing of conformally killing forms

In the present paper we consider the little-known Sampson operator that is
 strongly elliptic and self-adjoint second order differential operator acting
 on covariant symmetric tensors on Riemannian manifolds. First of all, we
 review the results on this operator. Then we consider the properties of the
 Sampson operator acting on one-forms and symmetric two-tensors. We study
 this operator using the analytical method, due to Bochner, of proving
 vanishing theorems for the null space of a Laplace operator admitting
 a Weitzenb?ck decomposition. Further we estimate operator?s lowest
 eigenvalue.

/pdf/on-the-sampson-laplacian-1ajtnbvvvd.pdf

On the Sampson Laplacian

/pdf/from-infinitesimal-harmonic-transformations-to-ricci-srdhs7erzo.pdf

Irina Tsyganok

Papers

Differential Geometry of Special Mappings

Conformal Killing $L^{2}-$forms on complete Riemannian manifolds with nonpositive curvature operator

Theorems of existence and of vanishing of conformally killing forms

On the Sampson Laplacian

From infinitesimal harmonic transformations to Ricci solitons