Let M~(n+1) be a complete Riemannian manifold with boundary Partial derivM =: B ≠ φ which is a union of smooth hypersurfaces. We can see the precise definition of manifolds with boundary as billiard tables in [17]. Let q ∈ B be an arbitrary point at which B is smooth and Q_q the symmetry with respect to T_qB, i.e., Q_q (w) = w - 2 < w, N(q) > N(q) for any w ∈ T_qM, where < • , • > is the Riemannian metric in M and N is the unit normal vector field to B pointing inward.
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