Rigidity of the Stable Norm on Tori

Rigidez de la norma estable sobre toros

Given a closed, orientable Riemannian manifold, we study the stable norm on the real homology groups. In particular, for n≥ 2 we prove that a Riemanniann-torus, which has the same stable norms as a flat n-torus on the first andn-1 homology groups, is in fact isometric to the flat torus.

Dada una variedad Riemanniana, cerrada y orientable, estudiamos la norma estable sobre sus grupos de homología real. En particular, para n≥ 2 demostramos que si un n-toro Riemanniano tiene normas estables iguales a las normas estables de un n-toro plano sobre el primer y n-1 grupos de homología; entonces es isométrico a dicho toro plano.

Rigidity of the Stable Norm on Tori Rigidez de la norma estable sobre toros OSVALDO OSUNA1

1Universidad Michoacana, Morelia, México. Email: osvaldo@ifm.umich.mx 

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Abstract

Given a closed, orientable Riemannian manifold, we study the stable norm on the real homology groups. In particular, for n≥ 2 we prove that a Riemanniann-torus, which has the same stable norms as a flat n-torus on the first andn-1 homology groups, is in fact isometric to the flat torus.

Key words: Stable norm, p-norm, Poincaré duality.

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2000 Mathematics Subject Classification: 53C23, 53D25, 53C24.

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Resumen

Dada una variedad Riemanniana, cerrada y orientable, estudiamos la norma estable sobre sus grupos de homología real. En particular, para n≥ 2 demostramos que si un n-toro Riemanniano tiene normas estables iguales a las normas estables de un n-toro plano sobre el primer y n-1 grupos de homología; entonces es isométrico a dicho toro plano.

Palabras clave: Norma estable, p-norma, dualidad de Poincaré.

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Texto completo disponible en PDF

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References

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(Recibido en julio de 2008. Aceptado en abril de 2010)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n1a02, 
    AUTHOR  = {Osuna, Osvaldo}, 
    TITLE   = {{Rigidity of the Stable Norm on Tori}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {1}, 
    pages   = {15-21} 
}