Abstract
In this chapter, we study special types of isometric group actions on compact Riemannian manifolds, namely those with low cohomogeneity. The cohomogeneity of an action is the codimension of its principal orbits, and can also be regarded as the dimension of its orbit space. Low cohomogeneity is an indication that there are few orbit types, and that the original space has many symmetries. In this situation, it is possible to study many geometric features that are not at reach in the general case. Throughout this chapter, we emphasize connections between the geometry and topology of manifolds with low cohomogeneity stemming from curvature positivity conditions, such as positive (\(\sec > 0\)) and nonnegative (\(\sec \geq 0\)) sectional curvature.
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