On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

E.D. Rodionov,O.P. Khromova

doi:10.14258/izvasu(2020)4-19

Abstract

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known. In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

Highlights

Manifold, and, in particular, the influence of the signOnoefof tsheecitmiopnoarltanctuprrvoabtleumres ofoRniemthaneniatnopgoeolomgeictrayl isstrtuhcetuprreobolfema Rofieemstaanbnliisahninmgacnoinfonldec. tOiofnspabrettiwcueleanr cimurpvoatrutraenacneditnhetthoepsoelosgtyuodfiaesRiiesmtahneniqanuemstainoinfolodf, atnhde, iinnflpuaretniccuelaro,fthδe-ipnifnlucehnicnegofotfheRsiigenmoafnsneciatinonaml ceutrrivcastuoref opnostihteivteopsoelcotgioicnaallstcruurcvtuarteuroef aonRietmheangneioamn emtraincifaonldd
RuOsnsiea)of the important problems of Riemannian рии является задача об установлении связей меж- geometry is the problem of establishing connections ду кривизной
В данной работе исследована функция δзащемленности секционной кривизны компактной связной группы Ли G с биинвариантной римановой метрикой и произвольной связностью с векторным кручением

Summary

Introduction

Manifold, and, in particular, the influence of the signOnoefof tsheecitmiopnoarltanctuprrvoabtleumres ofoRniemthaneniatnopgoeolomgeictrayl isstrtuhcetuprreobolfema Rofieemstaanbnliisahninmgacnoinfonldec. tOiofnspabrettiwcueleanr cimurpvoatrutraenacneditnhetthoepsoelosgtyuodfiaesRiiesmtahneniqanuemstainoinfolodf, atnhde, iinnflpuaretniccuelaro,fthδe-ipnifnlucehnicnegofotfheRsiigenmoafnsneciatinonaml ceutrrivcastuoref opnostihteivteopsoelcotgioicnaallstcruurcvtuarteuroef aonRietmheangneioamn emtraincifaonldd. Рошо известен ряд теорем римановой геометрии: теорема Адамара–Картана о полном односвязном римановом многообразии неположительной секционной кривизны, теорема М. В случае неположительной секционной кривизны хорошо известны результаты работ [6, 7]. Пусть (M, g) риманово многообразие, ∇g связность Леви-Чивиты, ∇ связность с векторным кручением, определяемая по формуле

Results

Conclusion