scholarly journals 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

2020 ◽  
pp. 117-120
Author(s):  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Topological Structure ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Sectional Curvature ◽  
Levi Civita Connection ◽  
Connected Lie Group

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].

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

scholarly journals A note on para-holomorphic Riemannian–Einstein manifolds

2016 ◽  
Vol 13 (09) ◽  
pp. 1650107
Author(s):  
Cristian Ida ◽  
Alexandru Ionescu ◽  
Adelina Manea
Keyword(s):  
Complex Manifold ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Complex Geometry ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Einstein Metric ◽  
Einstein Manifolds ◽  
Norden Metric ◽  
Norden Metrics

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

THE HOMOGENEOUS LIFT TO THE (1, 1)-TENSOR BUNDLE OF A RIEMANNIAN METRIC

2013 ◽  
Vol 10 (04) ◽  
pp. 1350006 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
HASSAN NASRABADI ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Differentiable Manifold ◽  
Riemannian Metric ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Geometric Objects ◽  
Levi Civita Connection ◽  
Tensor Bundle

By using a Riemannian metric on a differentiable manifold, the homogeneous lift metric is introduced on the (1, 1)-tensor bundle of the Riemannian manifold. Some geometric objects related to this metric, such as the Levi-Civita connection, Riemannian curvature tensor and sectional curvature are calculated. Also, a para-Nordenian structure on the (1, 1)-tensor bundle with this metric is constructed and interesting properties of this structure are studied.

scholarly journals On the number of diffeomorphism classes in a certain class of Riemannian manifolds

1985 ◽  
Vol 97 ◽  
pp. 173-192 ◽  
Author(s):  
Takao Yamaguchi
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Homotopy Types

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

Choice of Riemannian Metrics for Rigid Body Kinematics

1996 ◽  
Author(s):  
Miloš Žefran ◽  
Vijay Kumar ◽  
Christopher Croke
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Three Dimensions ◽  
Euclidean Group ◽  
Screw Motion ◽  
Screw Motions ◽  
Rigid Body Motions ◽  
Two Parameter

Abstract The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, SE(3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of SE(3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show that the metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannian metrics.

Curvature of K-contact Semi-Riemannian Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 401-412 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Lorentzian Manifold ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

CLASSES OF GRADIENT RICCI SOLITONS

2011 ◽  
Vol 08 (04) ◽  
pp. 783-796 ◽  
Author(s):  
GABRIEL BERCU ◽  
MIHAI POSTOLACHE
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Wide Class ◽  
Positive Function ◽  
Polar Coordinates ◽  
Ricci Solitons ◽  
Positive Sectional Curvature ◽  
Gradient Ricci Solitons ◽  
Research Article

We introduce a study of Riemannian manifold M = ℝ2 endowed with a metric of diagonal type of the form [Formula: see text], where g is a positive function, of C∞-class, depending on the variable x2 only. We emphasize the role of metric [Formula: see text] in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

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