ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 2, pp. 69–85.
Allerton Press, Inc., 2018.
Original Russian Text
E.I. Yakovlev and T.A. Gonchar, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 77–95.
Geometry and Topology of Some Fibered Riemannian Manifolds
E. I. Yakovlev
Nizhny Novgorod State University
pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
Received October 21, 2016
Abstract—We investigate a principal G-bundle with G-invariant Riemannian metric on its total
space. We derive formulas describing the Levi-Civita connection and curvatures in two-dimensional
directions. We obtain estimates of the inﬂuence of properties of sectional curvatures to topological
invariants of the bundle.
Keywords: principal bundle, G-connection, Riemannian manifold, Levi-Civita connection,
1. STATEMENT OF THE PROBLEM
Let ξ =(E,p,B,G) be a smooth principal bundle with projection p : E → B and structure group G.
We assume that G acts on E on the right. Assume also that the manifold E possesses a Riemannian
metric g invariant under the action R : E × G → E. This means that for any v ∈ E, a ∈ G and
X,Y ∈ T
E we have the equality
(Y )) = g(X,Y ). (1)
Fiber bundles are interesting by themselves and have numerous applications. In particular, Lorentz
manifolds of this kind are used in theoretical physics in constructing Kaluza–Klein type models.
Riemannian ﬁber bundles proved to be useful in studying the dynamics of gyroscopic systems with a
multivalued action functional (see, e.g., [1–3]).
The purpose of this paper is to solve the following problems.
Problem 1. Computation of the Levi-Civita connection and sectional curvatures for a Riemannian
Problem 2. An estimate of the eﬀect of properties of sectional curvature of the manifold (E,g) on the
topological invariants of the bundle
The solution of Problem 1 implies derivation of formulas in which the indicated quantities are
expressed in terms of geometric objects on the manifolds B, G,andE induced by the Riemannian
metric g. Therefore, for a more precise formulation, we describe the construction of these objects and
We denote by V
the vertical subspace of the tangent space T
E, consisting of vectors tangent to
the ﬁber G
(b), b = p(v),andbyH
the orthogonal complement to V
in the Euclidean space
). Thenby(1)therelationH : v → H
is a G-connection on E.Letω be a connection form
for H,andΩ be its curvature form.