Harmonic maps relative to α-connections on Hessian domains

28/08/2013
Auteurs : Keiko Uohashi
OAI : oai:www.see.asso.fr:2552:5133
DOI :

Résumé

Harmonic maps relative to α-connections on Hessian domains

Média

Voir la vidéo

Métriques

529
185
226.52 Ko
 application/pdf
bitcache://7ebace477036f84c22818c9562a93b0d3b2886ee

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors financier

logo_gdr-mia.png
logo_inria.png
image010.png
logothales.jpg

Sponsors logistique

logo-minesparistech.jpg
logo-universite-paris-sud.jpg
logo_supelec.png
Séminaire Léon Brillouin Logo
logo_cnrs_2.jpg
logo_ircam.png
logo_imb.png
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/2552/5133</identifier><creators><creator><creatorName>Keiko Uohashi</creatorName></creator></creators><titles>
            <title>Harmonic maps relative to α-connections on Hessian domains</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2013</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Wed 16 Oct 2013</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Fri 25 May 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">7ebace477036f84c22818c9562a93b0d3b2886ee</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>15066</version>
        <descriptions>
            <description descriptionType="Abstract"></description>
        </descriptions>
    </resource>
.

Harmonic maps relative to -connections on Hessian domains Keiko UOHASHI Tohoku Gakuin Univ. Miyagi 985-8537, JAPAN uohashi@tjcc.tohoku-gakuin.ac.jp 1 α Contents • -connection on a statistical manifold • Level surfaces of a Hessian domain • Map for conformal equivalence • Harmonic map and -connections • -affine harmonic map -comparison with previous study- 2 α α α -connection on a statistical manifold • : Statistical manifold : dual connection of : Levi-Civita connection Def : -connection for ( ) 3 α ),(),(),( ZYgZYgZYXg XX ∇′+∇= ∇′ ∇ LC ∇= ∇′+∇ 2 ∇α)(α ∇ R∈α ⇔ ∇′ − +∇ + =∇ 2 1 2 1)( ααα ),,( gM ∇ ⇔ Remark (1) , , (2) : statistical manifold : dual connection of (3) : Hessian domain flat statistical manifold where : domain in a real affine space : canonical affine connection : non-degenerate Riemannian metric 4 ∇=∇ )1( LC ∇=∇ )0( ∇′=∇ − )1( ),,( )( gM α ∇ ),,( ϕDdgD =Ω ⇒ 1+ ⊂Ω n A D ϕDdg = )( α− ∇ )(α ∇ : gradient vector field of on on : level surface of : canonical immersion E ~ )() ~ ,( XdEXg ϕ= ϕ Ω ⇔ EEdE ~ ) ~ ( 1− −= ϕ }0|{ ≠Ω∈=Ω po dp ϕ oM Ω⊂ Ω→Mx : ϕ 5 Level surfaces of a Hessian domain : affine immersion for : induced affine connection : affine fundamental form : statistical submanifold induced by ),( Ex ),,( EE gDM ),,( MM gDM EYXgYDYD E X E X ),(+= TMYX ∈, E D E g 6 Structures on a level surface ),,( ϕDdgD =Ω Then Structure 1 coincides with Structure 2 . Thm. is a 1-conformally flat (i.e., projectively flat with symmetric Ricci tensor) statistical submanifold. [Uohashi, Ohara, Fujii (2000)a] Thm. are -conformally equivalent statistical submanifolds. [Uohashi, Ohara, Fujii (2000)b] [Uohashi (2002)] ),,( EE gDM ),,( MM gDM ⇓ ),,( MM gDM 7 R MM gDM ∈α α )},,{( )( α : affine coordinate system on : dual affine coordinate system : gradient mapping Fact : (a map between two level surfaces on ) s.t. , , , , is -conf. eq. transformation. { }11 ,, +n xx  { }1 * 1 * ,, +nxx  ii x x ∂ ∂ −= ϕ ι** 1: +→Ω nAι ⇔ 8 Map for conformal equivalence 1+n A )ˆ,ˆ,(),,( )()( gDMgDM αα → Ω π ιπι λ e=ˆ )( ))(( p epe λλ = )ˆ(ˆ)()( Mpe p ιιλ ∈ Mp ∈ α Remark Above map is a projective transformation for the dual coordinate. 9 π Harmonic map and -connections Def : is a harmonic map relative to the Levi-Civita connections The tension field vanishes, i.e., Def : is a harmonic map relative to -connections 10 α π ⇔ )ˆ,ˆ,ˆ(),,( gMgM LCLC ∇→∇ 0))())((ˆ()( ** ≡ ∂ ∂ ∇− ∂ ∂ ∇= ∂ ∂ ∂ ∂ j LC x j LC x ij xx g ii πππτ π )ˆ,ˆ,ˆ(),,( )()( gMgM αα ∇→∇ ⇔ 0)}())((ˆ{)( )( ** )( ≡ ∂ ∂ ∇− ∂ ∂ ∇= ∂ ∂ ∂ ∂ j x j x ij xx g ii αα πππτ α Proposition For harmonic map relative to -connections, Difficult for direct calculation! We treat maps between two level surfaces with respect to -conformal equivalence. 11 α 0)ˆ)1()1(( 2 = ∂ ∂ ∂ ∂ Γ−+ ∂ ∂ Γ−− ∂∂ ∂ jik k ijji ij xxxxx g βδ γ δβ γγ ππ α π α π ⇓ ⇓ α Thm. If or is a constant function on , a map : is a harmonic map relative to .
By definition of “ -conformal equivalence” See GSI 2013 Proceedings. 12 2 2 + − −= n n α λ π M )ˆ,ˆ,(),,( )()( gDMgDM αα → )ˆ,,( )()( αα DDg α Remark For hamonicity of , (1) ; : non-constant function (2) ; : non-constant function (3) harmonic maps do not exist. 13 λ∃ 2=n π 0=α⇔ 3=n π λ∃ 01 <<− α⇒ 0,1 >−≤ αα ⇒ 14 15 Problem Find applications of non trivial harmonic maps relative to -connections .α -affine harmonic map -comparison with previous study- Def : is a -affine harmonic map Remark : affine harmonic map [Jost, Simsir (2009)] [Shima (1995)] : Harmonic map 16 α π ⇔ 0)ˆ)1(( 2 = ∂ ∂ ∂ ∂ Γ+ ∂ ∂ Γ−− ∂∂ ∂ jik k ijji ij xxxxx g βδ γ δβ γγ πππ α π )ˆ,ˆ,ˆ(),,( )()( gMgM αα ∇→∇ 1=α α 0=α ⇒ ⇒ π π αα References K.Uohashi, A.Ohara and T.Fujii; 1-conformally flat statistical submanifolds, Osaka J. math. 37 (2000), 501-507. K.Uohashi, A.Ohara and T.Fujii; Foliations and divergences of flat statistical manifolds, Hiroshima Math. J. 30 (2000), 403-414. K.Uohashi; On -conformal equivalence of statistical submanifolds, J. of Geom. 75 (2002), 179-184. K. Uohashi; Harmonic maps relative to -connections on statistical manifolds, Applied Sciences, Vol.14 (2012),82-88. K. Uohashi; A Hessian Domain Constructed with a Foliation by 1-Conformally Flat Statistical Manifolds, International Mathematical Forum, Vol. 7 (2012), no. 48, 2363 – 2371. 17 α α