اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRiemannian symmetries in flag manifolds
395
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold $SO(5)/SO(2)times SO(2) times SO(1)$ what are the conditions for a metric adapted to the $mathbb{Z}_2^2$-symmetric structure to be naturally reductive.
قيم البحث
اقرأ أيضاً
A {em 2-Riemannian manifold} is a differentiable manifold exhibiting a 2-inner product on each tangent space. We first study lower dimensional 2-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate t
o each 2-Riemannian manifold a unique torsion free compatible pseudoconnection. Using it we define a curvature for 2-Riemannian manifolds and study its properties. We also prove that 2-Riemannian pseudoconnections do not have Koszul derivatives. Moreover, we define stationary vector field with respect to a 2-Riemannian metric and prove that the stationary vector fields in $mathbb{R}^2$ with respect to the 2-Riemannian metric induced by the Euclidean product are the divergence free ones.
We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of $n+1$ p
oints on an $n$-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism between a neighbourhood of a Euclidean simplex, and a neighbourhood containing the points on the manifold.
We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the d
omains that are swept out by any given tractor/tractrix system. These expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we prove explicit estimates for them based on Rauchs and Toponogovs comparison theorems. Moreover, the general length shortening property of tractor/tractrix systems is used to generate geodesics in homotopy classes of curves in the ambient manifold.
We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riema
nnian manifolds this finite blocking property characterizes the flat metrics. Using entropy considerations, we verify this conjecture amongst metrics with nonpositive sectional curvatures. Using the same approach, K. Burns and E. Gutkin have independently obtained this result. Additionally, we show that compact quotients of Euclidean buildings have the finite blocking property. On the positive curvature side, we conjecture that compact Riemannian manifolds with the same blocking properties as compact rank one symmetric spaces are necessarily isometric to a compact rank one symmetric space. We include some results providing evidence for this conjecture.
We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted con
nection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern-Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns-Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
