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A generalization of the exterior product of differential forms combining Hom-valued forms

Christian Gross (1997)

Commentationes Mathematicae Universitatis Carolinae

This article deals with vector valued differential forms on C ∞ -manifolds. As a generalization of the exterior product, we introduce an operator that combines Hom ( ⨂ s ( W ) , Z ) -valued forms with Hom ( ⨂ s ( V ) , W ) -valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.

A generalized exponential map for an affinely homogeneous cone

Umberto Sampieri (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Dato un cono V aperto non vuoto, convesso, regolare e affinemente omogeneo in uno spazio vettoriale reale W di dimensione finita si prova che per ogni v appartenente a V esiste un diffeomorfismo E v : W → V che soddisfa le condizioni seguenti E1) E v ⁢ ( 0 ) = v ; E2) det ⁡ ( d ⁢ E v ⁢ ( y ) ) = Φ V ⁢ ( E v ⁢ ( y ) ) - 1 per ogni y appartenente a W ove Φ V : V → 𝐑 + è la funzione caratteristica di V .

A geometry on the space of probabilities (II). Projective spaces and exponential families.

Henryk Gzyl, Lázaro Recht (2006)

Revista Matemática Iberoamericana

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3 ) with constant ( r + 1 ) -th mean curvature H r + 1 immersed into the Euclidean sphere 𝕊 n + 1 , where r ∈ { 1 , ... , n - 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of 𝕊 n + 1 , a region that is determined by a totally umbilical sphere of 𝕊 n + 1 . We also provide a rigidity result for such hypersurfaces.

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