We examine the geometry of subspace homogeneous probability measures in terms 2-Wasserstein metric on space all Hilbert spaces functions. show that, appropriate spaces, geodesics joining stay and, as a result, themselves form nonpositive curvature sense A. D. Alexandrov.

The critical point equation arises as a of the total scalar curvature functional defined on space constant metrics unit volume compact manifold. In this equation, there exists function f manifold that satisfies following $$\begin{aligned} (1+f)\mathrm{Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. \end{aligned}$$ It has been conjectured if (g, f) is solution then g Einstein and so (M, g) isometric to s...

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the positive p-curvature and k-positive Ricci curvature a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided manifold admits such metric.