On the geometric meaning of the Bochner tensor

1975 ◽  
Vol 4 (1) ◽  
Author(s):  
DavidE. Blair
Keyword(s):  
Geometric Meaning ◽  
Bochner Tensor

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Variation formulas for principal functions, II: Applications to variation for harmonic spans

2011 ◽  
Vol 204 ◽  
pp. 19-56 ◽  
Author(s):  
Sachiko Hamano ◽  
Fumio Maitani ◽  
Hiroshi Yamaguchi
Keyword(s):  
Direct Application ◽  
Total Space ◽  
Geometric Meaning ◽  
Principal Function ◽  
Variation Formula ◽  
Circular Slit ◽  
Moving Domain

AbstractA domainD⊂ Czadmits the circular slit mappingP(z) fora, b∈Dsuch thatP(z) – 1/(z–a) is regular ataandP(b) = 0. We callp(z) =log|P(z)|theLi-principal functionandα= log |P′(b)| theL1-constant, and similarly, the radial slit mappingQ(z) implies theL0-principal functionq(z) and theL0-constantβ. We calls=α–βtheharmonic spanfor (D, a, b). We show the geometric meaning ofs. Hamano showed the variation formula for theL1-constantα(t) for the moving domainD(t) in Czwitht∈B:= {t∈ C: |t| <ρ}. We show the corresponding formula for theL0-constantβ(t) forD(t) and combine these to prove that, if the total spaceD =∪t∈B(t, D(t)) is pseudoconvex inB× Cz, thens(t) is subharmonic onB. As a direct application, we have the subharmonicity of log coshd(t) onB, whered(t) is the Poincaré distance betweenaandbonD(t).

scholarly journals Projective geometry in characteristic one and the epicyclic category

2015 ◽  
Vol 217 ◽  
pp. 95-132
Author(s):  
Alain Connes ◽  
Caterina Consani
Keyword(s):  
Projective Geometry ◽  
Free Module ◽  
The Self ◽  
Vector Spaces ◽  
Geometric Meaning ◽  
Absolute Point ◽  
One Dimensional ◽  
Self Duality ◽  
Finite Dimensional ◽  
The One

AbstractWe show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield ofmax-plus integersℤmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces arefiniteand provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

Vanishing g-Bochner Tensor of Viasman-Gray Manifold

2015 ◽  
Vol 8 (6) ◽  
pp. 470-481
Author(s):  
Habeeb Abood ◽  
Ibtehal Abed
Keyword(s):  
Bochner Tensor
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