scholarly journals A unified approach to compact symmetric spaces of rank one

2010 ◽  
Vol 118 (1) ◽  
pp. 43-87
Author(s):  
Adam Korányi ◽  
Fulvio Ricci
Keyword(s):  
Symmetric Spaces ◽  
Unified Approach ◽  
Rank One ◽  
Compact Symmetric Spaces

scholarly journals Homotopy of compact symmetric spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 221-228 ◽  
Author(s):  
John M. Burns
Keyword(s):  
Symmetric Spaces ◽  
Unified Approach ◽  
Topological Characterization ◽  
New Approach ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Compact Symmetric Spaces

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

scholarly journals A survey on Inverse mean curvature flow in ROSSes

2017 ◽  
Vol 4 (1) ◽  
pp. 245-262
Author(s):  
Giuseppe Pipoli
Keyword(s):  
Mean Curvature ◽  
Symmetric Spaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Unified Approach ◽  
Inverse Mean Curvature Flow ◽  
Rank One ◽  
Similarities And Differences ◽  
Mean Convex ◽  
Convex Hypersurfaces

AbstractIn this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.

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