In each of them we choose a k -dimensional submanifold NC and N−C , points p±C ∈ N±C and a chart of M±C around p±C given by Fermi coordinates such that all eigenvalues of the second fundamental form with respect to those coordinates at p±C are given by ±C (this is always possible, cf. [SpIV, Chapter 7]).
Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces
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