Any diffeomorphism-invariant, fiberwise-convex space of Riemannian metrics on the unit disk is contractible, covering positive Gauss curvature and convex or geodesic boundary conditions.
Carr, Construction of manifolds of positive scalar curvature , Trans
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Contractibility results for certain spaces of Riemannian metrics on the disc
Any diffeomorphism-invariant, fiberwise-convex space of Riemannian metrics on the unit disk is contractible, covering positive Gauss curvature and convex or geodesic boundary conditions.