English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let P(Mf G) be a principal fiber bundle with Lie structure group G and base M, with differentiability of class C00 for all manifolds and mappings involved in its definition. Given a connection of class C00 in P, the holonomy group <£(#) with reference point x£P is defined as the subgroup of G consisting of all elements a£G such that x and x-a can be joined by a horizontal curve in P. Here, a horizontal curve usually means either a piecewise C^-curve with horizontal tangent vec­ tors or a piecewise C°°-curve with horizontal tangent vectors. In fact, for any specified degree of differentiabi lity k, l^k^ <*>, we can use piecewise Cfc-curves exclusively thus defining the holonomy group $k(x) which depends presumably on k. Of course, it is evident that *i(x)3*2(x)D • • • D*oo(x). We asked ourselves whether these holonomy groups are the same or not, and, as far as we know, there has not been any record concern­ ing this question. One might think that approximation of piecewise O-curves by piecewise C°°-curves will settle this question, but this method does not seem to work too easily in view of the fact that a Lie group can admit a Lie subgroup of lower dimension which is every­ where dense. We present here a proof of THEOREM. 4>I(X) =<£>00(X). We follow the terminologies of (2) in which piecewise C^-curves are used exclusively. Let P(x) be the set of all points in P which can be joined to x by a horizontal piecewise C°°-curve. It is known that P(x) is a subbundle of P with structure group ^(x) (2, p. 37). We define a distribution A on P by Ax= Tx(P(x)) for each x£P. We can prove that A is differentiate. It is involutive, since for each point x of P, P(x) is an integral manifold of A through x. It is also easy to show that P(x) is indeed a maximal integral manifold of A through x. (Remark that the distribution A considered in (2, p. 39) is the restric­ tion of our distribution A here to P(x) for a fixed x.) Now let a£$i(x). This means that there is a horizontal piecewise C^-curve x(t) such that

On the degree of differentiability of curves used in the definition of the holonomy group