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We characterize isometric vector fields through a distance resembling the Gromov–Hausdorff one (Burago D, Burago Y, Ivanov S in A course in metric geometry. Graduate studies in mathematics, vol 33, American Mathematical Society, Providence, 2001). We then use this distance to generate a metrizable topology in the space of vector fields on closed manifolds up to isometry. We prove that the set of Killing vector fields is closed with respect to this topology. [ABSTRACT FROM AUTHOR]

We present Low Distortion Local Eigenmaps (LDLE), a manifold learning technique which constructs a set of low distortion local views of a data set in lower dimension and registers them to obtain a global embedding. The local views are constructed using the global eigenvectors of the graph Laplacian and are registered using Procrustes analysis. The choice of these eigenvectors may vary across the regions. In contrast to existing techniques, LDLE can embed closed and non-orientable manifolds into their intrinsic dimension by tearing them apart. It also provides gluing instruction on the boundary of the torn embedding to help identify the topology of the original manifold. Our experimental results will show that LDLE largely preserved distances up to a constant scale while other techniques produced higher distortion. We also demonstrate that LDLE produces high quality embeddings even when the data is noisy or sparse. [ABSTRACT FROM AUTHOR]

Abstract: Given positive integers p and q, a -solid torus is a manifold diffeomorphic to while a -torus in a closed manifold M is the image of a differentiably embedding . We prove that if with or , then M is homeomorphic to whenever every -torus bounds a -solid torus. We also prove for that every closed n-manifold for which every -torus bounds an irreducible manifold is irreducible. Consequently, every closed 3-manifold for which every torus bounds an irreducible manifold is irreducible. [Copyright &y& Elsevier]

Let ( M,g( t)), 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| ≤ C/ t for some constant C > 0 and g( t) evolving by the Ricci flow . In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type on M × (0, T], where a > 0 and b ∈ ℝ. [ABSTRACT FROM AUTHOR]

Abstract: Instead of Lusternik–Schnirelmann -category we define the -cohomological dimension by the cohomology with local coefficients and a gap theorem for closed manifolds is shown to hold for this invariant. [Copyright &y& Elsevier]