17 research outputs found
Strong topology on the set of persistence diagrams
We endow the set of persistence diagrams with the strong topology (the
topology of countable direct limit of increasing sequence of bounded subsets
considered in the bottleneck distance). The topology of the obtained space is
described.
Also, we prove that the space of persistence diagrams with the bottleneck
metric has infinite asymptotic dimension in the sense of Gromov.Comment: 6 page
There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics
We prove the following statement: Let g be a light-line-complete
pseudo-Riemannian Einstein metric of indefinite signature on a connected
(n>2)-dimensional manifold M. Assume that a conformally equivalent metric is
also Einstein. Then, the metrics are proportional with a constant coefficient.
If in addition the manifold is closed, the assumption that the manifold is
light-line-complete could be omitted.Comment: 3 pages, no figure
A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics
We generalize the following classical result of Fubini to pseudo-Riemannian metrics: if three essentially different metrics on an (n ≥ 3)-dimensional manifold M share the same unparametrized geodesics, and two of them (say, g and g) are strictly nonproportional (that is, the minimal polynomial of the g-self-adjoint (1, 1)-tensor defined by g coincides with the characteristic polynomial) at least at one point, then they have constant sectional curvature
Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
We prove an important partial case of the pseudo-Riemannian version of the
projective Lichnerowicz conjecture stating that a complete manifold admitting
an essential group of projective transformations is the round sphere (up to a
finite cover).Comment: 32 pages, one .eps figure. The version v1 has a misprint in Theorem
1: I forgot to write the assumption that the degree of mobility is greater
than two. The versions v3, v4 have only cosmetic changes wrt v
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