On the Construction of Isospectral Manifolds

2004 ◽  
pp. 1-14
Author(s):  
Werner Ballmann
Keyword(s):  
Isospectral Manifolds

Riemannian Coverings and Isospectral Manifolds

1985 ◽  
Vol 121 (1) ◽  
pp. 169 ◽  
Author(s):  
Toshikazu Sunada
Keyword(s):  
Isospectral Manifolds

COMPACTNESS OF D-ISOSPECTRAL METRICS

2008 ◽  
Vol 05 (01) ◽  
pp. 49-61 ◽  
Author(s):  
HYUNSUK KANG
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Gauge Equivalence ◽  
Isospectral Manifolds

We generalise the compactness of isospectral metrics in the C∞-topology modulo gauge equivalence in dimension three to the case for Schrödinger operators with bounded potentials, as well as the finiteness of diffeomorphism types of isospectral manifolds.

Survey of Isospectral Manifolds

2000 ◽  
pp. 747-778 ◽  
Author(s):  
Carolyn S Gordon
Keyword(s):  
Isospectral Manifolds

scholarly journals Isospectral drums and simple groups

2018 ◽  
Vol 15 (04) ◽  
pp. 1850060
Author(s):  
Koen Thas
Keyword(s):  
Physical Properties ◽  
The Other ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Planar Domains ◽  
Other Hand ◽  
Isospectral Manifolds ◽  
Transplantation Technique ◽  
Rule Out

Nearly every known pair of isospectral but nonisometric manifolds — with as most famous members isospectral bounded [Formula: see text]-planar domains which makes one “not hear the shape of a drum” [M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73(4 part 2) (1966) 1–23] — arise from the (group theoretical) Gassmann–Sunada method. Moreover, all the known [Formula: see text]-planar examples (so counter examples to Kac’s question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassmann–Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the known planar examples). To that end, we introduce four properties — called FF, MAX, PAIR and INV — inspired by natural physical properties (which rule out trivial constructions), that are satisfied for each of the known planar examples. Vice versa, we show that length equivalent manifolds with FF, MAX, PAIR and INV which arise from the Gassmann–Sunada method, must fall under one of our prior constructions, thus describing a precise classification of these objects. Due to the nature of our constructions and properties, a deep connection with finite simple groups occurs which seems, perhaps, rather surprising in the context of this paper. On the other hand, our properties define in some sense physically irreducible pairs of length equivalent manifolds — “atoms” of general pairs of length equivalent manifolds, in that such a general pair of manifolds is patched up out of irreducible pairs — and that is precisely what simple groups are for general groups.

scholarly journals Toda flows and isospectral manifolds

1992 ◽  
Vol 115 (3) ◽  
pp. 837-837 ◽  
Author(s):  
Leonid Faybusovich
Keyword(s):  
Isospectral Manifolds

Construction of Isospectral Manifolds

2011 ◽  
Author(s):  
Manlio Bordoni ◽  
Carlos Herdeiro ◽  
Roger Picken
Keyword(s):  
Isospectral Manifolds

scholarly journals Towers of isospectral manifolds

2009 ◽  
Vol 54 (1) ◽  
pp. 1-7
Author(s):  
Manlio Bordoni ◽  
Verita Marino
Keyword(s):  
Isospectral Manifolds

scholarly journals Constructing Isospectral But Non-Isometric Riemannian Manifolds

1992 ◽  
Vol 35 (3) ◽  
pp. 303-310 ◽  
Author(s):  
Sheng Chen
Keyword(s):  
Riemannian Manifolds ◽  
Isospectral Manifolds

AbstractIn this paper we examine the examples of isospectral but non-isometric Riemannian manifolds given by Milnor, Ikeda, and Vignéras. Of these, only Milnor's example is accounted for by Sunada's method of constructing isospectral manifolds, and even then only as an "unnatural" construction.

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