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Nilpotent structures and invariant metrics on collapsed manifolds. (English) Zbl 0758.53022
Let M be a complete Riemannian n-manifold of bounded curvature. For any ε>0, the ε-collapsed part of M is defined as the set 𝒞 n (ε) of points at which the injectivity radius of the exponential map is <ε. The authors study the structure of the ε-collapsed part of a manifold M for suitably small ε. The main results show that the local geometry of 𝒞 n (ε) is encoded partially in the symmetry properties of a nearby metric. More precisely, a given metric can be closely approximated by one that admits a sheaf of nilpotent Lie algebras of local Killing vector fields pointing in all sufficiently collapsed directions of 𝒞 n (ε). This sheaf is called the nilpotent Killing structure. A detailed construction of this nilpotent Killing structure is presented, its properties are established and some applications to the description of collapses of a Riemannian manifold M are indicated.

MSC:
53C20Global Riemannian geometry, including pinching