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We present a new method for the study of general higher dimensional Kaluza-Klein theories. Our
new approach is based on the Riemannian adapted connection and on a theory of adapted tensor fields in the
ambient space. We obtain, in a covariant form, the fully general 4D equations of motion in a (4 +

As it is well known, by the Kaluza-Klein theory, the unification of Einstein’s theory of general relativity with Maxwell’s theory of electromagnetism was achieved. In a modern terminology, this theory is developed on a trivial principal bundle over the usual

Two conditions have been imposed in the classical Kaluza-Klein theory and in most of the above generalizations: the “cylinder condition” and the “compactification condition.” The former condition assumes that all the local components of the pseudo-Riemannian metric on the ambient space do not depend on the extra dimensions, while the latter requires that the fibre must be a compact manifold.

In 1938, Einstein and Bergmann [

Recently, we presented a new point of view on a general Kaluza-Klein theory in a

The present paper is the first in a series of papers devoted to the study of general Kaluza-Klein theory with arbitrary gauge group. More precisely, our approach is developed on a principal bundle

The whole study is based on the Riemannian adapted connection that we construct in this paper and on a

Now, we outline the content of the paper. In Section

Let

respectively.

Throughout the paper we use the ranges of indices:

Next, from (

As we apply the above objects to physics, we need a coordinate presentation for them. First, we recall (cf. [

By a different method, the above Yang-Mills fields have been first introduced by Cho [

Next, we express the pseudo-Riemannian metric

Finally, we consider two coordinate systems

In the present section we develop a tensor calculus on

First, we consider the dual vector bundles

Next, we will construct some adapted tensor fields which are deeply involved in our study. First, we denote by

for all

We close this section with a local presentation of the adapted tensor fields

If in particular

In all the papers published so far on Kaluza-Klein theories with nonabelian gauge group, the local components

In a previous paper (cf. [

First, we denote by

Next, we say that

Let

First, define

As

It is important to note that both

Throughout the paper, all local components for linear connections and adapted tensor fields are defined with respect to the adapted frame field

Next, we consider

The Levi-Civita connection

According to decomposition (

In this section we present the first achievement of the new method which we develop on general

Let

The equations of motion in a general gauge Kaluza-Klein space

We call (

Next, we suppose that only (

In this section we show that the set of geodesics in

The study of geodesics of

(i) A curve

(ii) A curve

It is noteworthy that the equations in (

Coming back to the general case, we say that a curve

A curve

A similar characterization can be given for vertical geodesics in

Next, we consider the case of the integrable horizontal distribution; that is, (

Now, we say that

An oblique geodesic of

Next, we say that

In the present paper we obtain, for the first time in the literature, the fully general equations of motion in a general gauge Kaluza-Klein space (cf. (

The method developed in the present paper opens new perspectives in the study of some other important concepts from higher dimensional physical theories. Here we have in mind an approach of the dynamics in such spaces under the effect of an extra force whose existence is guaranteed by the extra dimensions. In a particular case (see Section

All these problems deserve further studies which might show how far the concepts induced by the extra dimensions can be related to the real matter.