Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature
In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature (Shen in Differential Geometry of Spray and Finsler Spaces, p. 184, 2001 ). In this paper we provide an answer to this question within the class of Finsler metrics of scalar...
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| description | In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature (Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
). In this paper we provide an answer to this question within the class of Finsler metrics of scalar flag curvature. In Theorem
3.1
, we show that the answer is negative, for non-vanishing scalar flag curvature (SFC). The answer is known to be positive when the SFC vanishes (Grifone and Muzsnay in Variational Principles for Second-Order Differential Equations,
2000
; Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
), and this positive answer is related to the existence of many solutions to Hilbert’s Fourth Problem. As a generalization of this problem, we can ask if it is possible for a given spray, with non-vanishing SFC, to represent, after reparameterization, the geodesic spray of a Finsler metric. In Proposition
3.3
, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature. |
| doi_str_mv | 10.1007/s12220-015-9661-z |
| format | Article |
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2001
). In this paper we provide an answer to this question within the class of Finsler metrics of scalar flag curvature. In Theorem
3.1
, we show that the answer is negative, for non-vanishing scalar flag curvature (SFC). The answer is known to be positive when the SFC vanishes (Grifone and Muzsnay in Variational Principles for Second-Order Differential Equations,
2000
; Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
), and this positive answer is related to the existence of many solutions to Hilbert’s Fourth Problem. As a generalization of this problem, we can ask if it is possible for a given spray, with non-vanishing SFC, to represent, after reparameterization, the geodesic spray of a Finsler metric. In Proposition
3.3
, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature.</description><identifier>ISSN: 1050-6926</identifier><identifier>EISSN: 1559-002X</identifier><identifier>DOI: 10.1007/s12220-015-9661-z</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Convex and Discrete Geometry ; Curvature ; Differential equations ; Differential Geometry ; Dynamical Systems and Ergodic Theory ; Flags ; Fourier Analysis ; Geometry ; Global Analysis and Analysis on Manifolds ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Variational principles</subject><ispartof>The Journal of Geometric Analysis, 2016-10, Vol.26 (4), p.3056-3065</ispartof><rights>Mathematica Josephina, Inc. 2015</rights><rights>COPYRIGHT 2016 Springer</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c355t-112d657d1812f2dfe41461aef8a52311b29c33ceb5f21c4631690237a370e1353</citedby><cites>FETCH-LOGICAL-c355t-112d657d1812f2dfe41461aef8a52311b29c33ceb5f21c4631690237a370e1353</cites><orcidid>0000-0002-8506-7567</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12220-015-9661-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12220-015-9661-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Bucataru, Ioan</creatorcontrib><title>Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature</title><title>The Journal of Geometric Analysis</title><addtitle>J Geom Anal</addtitle><description>In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature (Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
). In this paper we provide an answer to this question within the class of Finsler metrics of scalar flag curvature. In Theorem
3.1
, we show that the answer is negative, for non-vanishing scalar flag curvature (SFC). The answer is known to be positive when the SFC vanishes (Grifone and Muzsnay in Variational Principles for Second-Order Differential Equations,
2000
; Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
), and this positive answer is related to the existence of many solutions to Hilbert’s Fourth Problem. As a generalization of this problem, we can ask if it is possible for a given spray, with non-vanishing SFC, to represent, after reparameterization, the geodesic spray of a Finsler metric. In Proposition
3.3
, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature.</description><subject>Abstract Harmonic Analysis</subject><subject>Convex and Discrete Geometry</subject><subject>Curvature</subject><subject>Differential equations</subject><subject>Differential Geometry</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Flags</subject><subject>Fourier Analysis</subject><subject>Geometry</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Variational principles</subject><issn>1050-6926</issn><issn>1559-002X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kFFLwzAQx4soqNMP4FvB5867pEnbxzGdCgMFFXyLMb2Mzq6ZSTvQT29GffBFDnKXu_8vOf5JcoEwRYDiKiBjDDJAkVVSYvZ9kJygEFUGwF4PYw0CMlkxeZychrAGyCXPi5PkbTF0H2k8TN-4LqS6q9NH79YU7ztKr8k6v9HjzNn0aev116haNF1oyceWNjQOjW61TxetXqXzwe90P3g6S46sbgOd_-ZJ8rK4eZ7fZcuH2_v5bJkZLkSfIbJaiqLGEplltaUcc4mabKkF44jvrDKcG3oXlqGJy6OsgPFC8wIIueCT5HJ8d-vd50ChV2s3-C5-qbAsoSzyCvaq6aha6ZZU01nXe21i1LRpjOvINrE_K6DgvBJCRgBHwHgXgiertr7ZaP-lENTeeTU6r6Lzau-8-o4MG5kQtd2K_J9V_oV-AExEhWQ</recordid><startdate>20161001</startdate><enddate>20161001</enddate><creator>Bucataru, Ioan</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>IAO</scope><orcidid>https://orcid.org/0000-0002-8506-7567</orcidid></search><sort><creationdate>20161001</creationdate><title>Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature</title><author>Bucataru, Ioan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c355t-112d657d1812f2dfe41461aef8a52311b29c33ceb5f21c4631690237a370e1353</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Convex and Discrete Geometry</topic><topic>Curvature</topic><topic>Differential equations</topic><topic>Differential Geometry</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Flags</topic><topic>Fourier Analysis</topic><topic>Geometry</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Variational principles</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bucataru, Ioan</creatorcontrib><collection>CrossRef</collection><collection>Gale Academic OneFile</collection><jtitle>The Journal of Geometric Analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bucataru, Ioan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature</atitle><jtitle>The Journal of Geometric Analysis</jtitle><stitle>J Geom Anal</stitle><date>2016-10-01</date><risdate>2016</risdate><volume>26</volume><issue>4</issue><spage>3056</spage><epage>3065</epage><pages>3056-3065</pages><issn>1050-6926</issn><eissn>1559-002X</eissn><abstract>In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature (Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
). In this paper we provide an answer to this question within the class of Finsler metrics of scalar flag curvature. In Theorem
3.1
, we show that the answer is negative, for non-vanishing scalar flag curvature (SFC). The answer is known to be positive when the SFC vanishes (Grifone and Muzsnay in Variational Principles for Second-Order Differential Equations,
2000
; Shen in Differential Geometry of Spray and Finsler Spaces, p. 184,
2001
), and this positive answer is related to the existence of many solutions to Hilbert’s Fourth Problem. As a generalization of this problem, we can ask if it is possible for a given spray, with non-vanishing SFC, to represent, after reparameterization, the geodesic spray of a Finsler metric. In Proposition
3.3
, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12220-015-9661-z</doi><tpages>10</tpages><orcidid>https://orcid.org/0000-0002-8506-7567</orcidid></addata></record> |
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| language | eng |
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| source | SpringerNature Complete Journals |
| subjects | Abstract Harmonic Analysis Convex and Discrete Geometry Curvature Differential equations Differential Geometry Dynamical Systems and Ergodic Theory Flags Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Mathematical analysis Mathematics Mathematics and Statistics Variational principles |
| title | Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature |
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