The Banach manifold C(M,N)

2019 ◽  
Vol 63 ◽  
pp. 166-185 ◽  
Author(s):  
Johannes Wittmann
Keyword(s):  
Author(s):  
Enrique Covarrubias

The main contribution of this paper is to place smooth infinite economies in the setting of the equilibrium manifold and the natural projection map à la Balasko. We show that smooth infinite economies have an equilibrium set that has the structure of a Banach manifold and that the natural projection map is smooth. We define regular and critical economies, and regular and critical prices, and we show that the set of regular economies coincides with the set of economies whose excess demand function has only regular prices. Generic determinacy of equilibria follows as a by-product.


2015 ◽  
Vol 12 (07) ◽  
pp. 1550072 ◽  
Author(s):  
Pradip Mishra

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.


1997 ◽  
Vol 17 (3) ◽  
pp. 531-564 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the ${\cal C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the conjugacy (with a different extension) is in fact already ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal C}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.


2017 ◽  
Vol 17 (2) ◽  
pp. 175-189 ◽  
Author(s):  
Ali Suri

AbstractThe tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.


2002 ◽  
Vol 31 (6) ◽  
pp. 375-379
Author(s):  
El-Said R. Lashin ◽  
Tarek F. Mersal

We prove that an essential hypersurface of second order in an infinite dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature.


Author(s):  
S. C. P. Halakatti ◽  
Archana Halijol ◽  

2004 ◽  
Vol 2004 (2) ◽  
pp. 99-104
Author(s):  
El-Said R. Lashin ◽  
Tarek F. Mersal

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse, in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi-Riemannian Banach space is an essential hypersurface of second order.


2004 ◽  
Vol 11 (01) ◽  
pp. 71-77 ◽  
Author(s):  
R. F. Streater

Let [Formula: see text] be a separable Hilbert space. We consider the manifold [Formula: see text] consisting of density operators ρ on [Formula: see text] such that ρp is of trace class for some p ɛ (0, 1). We say [Formula: see text] is nearby ρ if there exists C > 1 such that C−1 ρ < σ < Cρ. We show that the space of nearby points to ρ can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish [Formula: see text] with a norm making it into a Banach manifold.


2015 ◽  
Vol 17 (04) ◽  
pp. 1550016 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.


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