The paper proves in all dimensions that the total curvature inequality implies the isoperimetric inequality in Cartan-Hadamard manifolds, and establishes a new comparison formula for total curvature of level sets.
Remarks on manifolds with two sided curvature bounds
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abstract
We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
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Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds
The paper proves in all dimensions that the total curvature inequality implies the isoperimetric inequality in Cartan-Hadamard manifolds, and establishes a new comparison formula for total curvature of level sets.