2017 Fiscal Year Final Research Report
Geometry of barycenter map on Hadamard manifolds admitting Busemann-Poisson kernel
| Project/Area Number |
15K17545
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Nippon Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Keywords | アダマール多様体 / 確率測度の空間 / フィッシャー計量 / 重心 / 幾何平均 / 測地線 / 調和多様体 / 球フーリエ変換 |
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| Outline of Final Research Achievements |
(1) We defined the normalized geometric mean of two positive probability measures. By using this notion, we found that there exists the unique geodesic segment joining arbitrary two probability measures. Moreover, we showed that a geodesic segment belongs entirely to a fiber of the barycenter map on a Hadamard manifold, if and only if endpoints of the geodesic and its normalized geometric mean belong same fiber.
(2) We obtained some properties of geodesics with respect to alpha-connection on the space of all probability measures with positive density function.
(3) We defined a class of Harmonic manifolds of hypergeometric type and we developed the theory of the spherical Fourier transform on a Hadamard harmonic manifold which is of hypergeometric type.
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| Free Research Field |
微分幾何学
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