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Summary: GAUSS EQUATION AND INJECTIVITY RADII FOR
SUBSPACES
IN SPACES OF CURVATURE BOUNDED ABOVE
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. A Gauss Equation is proved for subspaces of Alexandrov spaces
of curvature bounded above by K. That is, a subspace of extrinsic curvature
A, de ned by a cubic inequality on the di erence of arc and chord, has
intrinsic curvature K +A 2 . Sharp bounds on injectivity radii of subspaces,
new even in the Riemannian case, are derived.
1. Introduction
Alexandrov spaces are metric spaces with curvature bounds in the sense of local
triangle comparisons with constant curvature spaces. In this paper, we consider
spaces of curvature bounded above (CBA), and their global counterparts, CAT(K)
spaces. Examples include Riemannian manifolds with upper sectional curvature
bounds, possibly with boundary ([ABB1]), polyhedra with link conditions, and Tits
boundaries and asymptotic cones (see [BH]). A key property of CAT(K) spaces
is their preservation under Gromov-Hausdor convergence. CAT(K) spaces are
appropriate target spaces in harmonic map theory (see, for example, [EF, GS, J]),
and play an important role in geometric group theory (see [BH]).
Analogues of the Gauss Equation, governing the passage of curvature bounds to
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