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甬江數學講壇75講（2020年第2講）

發布日期：2020-01-10 作者：數學與統計學院 文章來源：未知  責任編輯：

人：莫小歡

報告時間：2020114 1000-1100

報告地點：龍賽理科樓116會議室

專家簡介：莫小歡, 北京大學數學科學學院教授，博士生導師，長期從事幾何學的教學和研究。2002年榮獲教育部提名國家自然科學獎一等獎（獨立），2007年主持的《幾何學》課程被評為國家級精品課，2009年獲得國家教學成果二等獎。先后應邀前往美國麻省理工大學，加州大學伯克利分校，德國馬克思·普朗克數學研究所（波思與萊比錫），法國高等科學研究院，意大利國際理論物理中心，巴西巴西利亞大學和坎皮納斯大學等世界著名科研機構訪問。2002年來連續主持國家自然科學基金項目.目前已發表學術論文110余篇，其中被SCI收錄90余篇，論文被引用達到527次。

報告題目-1Inverse problem of sprays with scalar curvature

報告摘要：Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. In this lecturewe discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray $G$ on a manifold is of vanishing $H$-curvature, but $G$ has not isotropic curvature, then $G$ is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain $U\subset\mathbb{R}^n$ with scalar curvature and vanishing $H$-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

報告題目-2Funk functions and constructions of dually flat Finsler metrics

報告摘要：Dually flat Finsler metrics arise from $\alpha$-flat information structures on Riemann-Finsler manifolds. Inspired by the theory of Funk functions and Hamel functions due to Li-Shen, we give a new approach to produce dually flat Finsler metrics in this lecture. Moreover, we manufacture new dually flat spherically symmetric Finsler metrics by using the standard Euclidean norm on $\mathbb{R}^n$.

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