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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 $\epsilon >0$, the $\epsilon$-collapsed part of $M$ is defined as the set ${𝒞}^{n}\left(\epsilon \right)$ of points at which the injectivity radius of the exponential map is $<\epsilon$. The authors study the structure of the $\epsilon$-collapsed part of a manifold $M$ for suitably small $\epsilon$. The main results show that the local geometry of ${𝒞}^{n}\left(\epsilon \right)$ 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 ${𝒞}^{n}\left(\epsilon \right)$. 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:
 53C20 Global Riemannian geometry, including pinching