Geometric Perturbation Theory for a Class of Fiber Bundles

Geometric Perturbation Theory for a Class of Fiber Bundles

Abstract

A systematic study of small, time-dependent, perturbations to geometric wave-equation domains is hardly existent. Acoustic enclosures are typical examples featuring locally reacting surfaces that respond to a pressure gradient or a pressure difference, alter the enclosure’s volume and, hence, the boundary conditions, and do so locally through their vibrations. Accordingly, the Laplace-Beltrami operator in the acoustic wave equation lives in a temporally varying domain depending on the displacement of the locally reacting surface from equilibrium. The resulting partial differential equations feature nonlinearities and are coupled though the time-dependent boundary conditions. The solution to the afore-mentioned problem, as presented here, integrates techniques from differential geometry, functional analysis, and physics. The appropriate space is shown to be a (perturbation) fiber bundle. In the context of a systematic perturbation theory, the solution to the dynamical problem is obtained from a combination of semigroup techniques for operator evolution equations and metric perturbation theory as used in AdS/CFT. Duhamel’s principle then yields a time-dependent perturbation theory, called geometric perturbation theory. It is analogous to, though different from, both Dirac’s time-dependent perturbation theory and the Magnus expansion. Specifically, the formalism demonstrates that the stationary-domain approximation for vibrational acoustics only introduces a small error. Analytic simplifications methods are derived in the framework of the piston approximation. Globally reacting surfaces (so-called pistons) replace the formerly locally reacting surfaces and reduce the number of independent variables in the underlying partial differential equations. In this way, a straightforwardly applicable formalism is derived for scalar wave equations on time-varying domains.

1 Introduction

The motivation of the present paper stems from a concrete and rather typical example originating from acoustics. The mathematical algorithm leading to high-precision perturbative solutions has been explained in details elsewhere [42] and is essential to mathematically understanding azimuthal sound localization in more than half of the terrestrial vertebrates [95]. What we do here is providing the foundations in terms of fiber-bundle theory.

As shown by Figure 1, both frogs and lizards and birds and, hence, more than half of the terrestrial vertebrates have left and right eardrums that are connected by an air-filled, interaural, cavity in between. This setup realizes the notion of internally coupled ears (ICE). Land-living vertebrates perform azimuthal sound by neuronally determining the time difference between left and right eardrum and what they measure at the eardrums is a key to understanding the ensuing auditory processing.

Let us assume that the -axis is through the center of and orthogonal to the two parallel eardrums and that the latter are positioned at and . An external sound source, which is usually far (enough) away from the two ears and depending on the time , generates a time-varying pressure, which is uniform at the eardrums and represented by and . The latter bring the eardrums into motion. Sound, loud as it may seem, leads to eardrum motion with extremely small amplitudes (nm). On the other hand, being of the order of cm, the perturbed dynamics seems, and is [42], accessible to perturbation theory.

To localize a sound source, the auditory system uses the time difference between left and right eardrum. Given the interaural distance , this time difference equals where is straight ahead. In passing we note that there is a degeneracy but that is a universal problem and not the issue here.

Figure 1: Schematic representation of Internally Coupled Ears (ICE) in frogs (a), lizards (b), and birds (c). The bird in (c) is seen from the top, the other two in (a) and (b) show a cross section, and all three exhibit the interaural cavity as a gray tube. Figures 1 (a)–(c) have been taken from van Hemmen et al. [95]

Because of the internal cavity with pressure , which effectively couples the two eardrums, the actual force on each of them is in fact the difference . The resulting dynamical system consists in the present case of a system of three, coupled, partial differential equations,

(1)
(2)

The former is the wave equation for in the three-dimensional interaural cavity, which up to the two fluctuating eardrums is fixed. The latter two equations labeled by refer to the two-dimensional damped wave equations for left and right eardrums with Laplacian , deviations parallel to the -axis, damping constant , as material constant characterizing the tympanic membranes, and stands for a at and a at . The other material constants can be found elsewhere [42]. The eardrums constitute through the time-dependent part of the two-dimensional manifolds that are the boundary and, hence, provide the time-dependent boundary conditions for the cavity’s three-dimensional Laplacian , where succinctly indicates this time dependence. The nature of will soon be analyzed in detail.

As for the boundary conditions themselves, they are no-slip [93, chapter 2], which means that the velocity of the membrane equals that of the air attached to it. We start here with the (linearized) Navier-Stokes equation without viscosity (), which is Euler, and require

(3)

Here is the density of air and denotes the outward unit normal vector to the boundary of the cavity with and as the acoustic fluid velocity, viz., air, so that is the normal component of . Because of the no-slip boundary condition [93, chapter 2], we can take the normal velocity at both eardrums equal to . That is, in our concrete example. Except for the possibly swinging eardrums, we have fixed boundaries for (1) so that there we are left with Neumann boundary conditions as . Since the borders of the eardrums are taken to be fixed, the membrane deflections in (2) satisfy Dirichlet boundary conditions on the borders of what we take here to be circular sectors: on .

Sloppily formulated, by putting the deflection as it was so small but keeping nonzero (3) on , one arrives at the acoustic boundary conditions introduced (ABC) by Beale and Rosencranz [6, 4]. What we do here is analyzing in full mathematical detail and generalizing the combination of (3) with the dynamics (D) incorporated by (1) & (2). That is, we generalize the acoustic boundary-condition dynamics (ABCD) that has been introduced elsewhere [42] to far more general manifolds and in its mathematically natural context of fiber bundles.

First summary

The above arguments show the need for combining the acoustic boundary conditions (ABC) with their dynamics (D). That is, symbolically

Here we present a formalism based on the combination of geometrical and analytic methods in the context of vibrational acoustics for models similar to the model of internally coupled ears. To this end, we first skim through the background of the approach that is going to appear.

Our problem

Our problem can be described in non-technical manner as follows. Suppose, you are given a suitably well-behaved volume with flexible boundaries. Suppose further that there is a small demon inside the volume that enacts by emission of a pressure wave on each point of the boundary from the interior a force density, i.e., pressure, if and only if you enact from the exterior of the boundary a force density, i.e., pressure, on that point on the boundary by emission of a (plane) wave. The boundaries start to vibrate due to the net difference of force per unit area enacted on the boundary from the exterior and interior. The overall question is on how the pressure in the interior of the enclosed volume evolves in time assuming that only the vibrations of the boundaries trigger the demon to emit a pressure wave?

In order to answer the above question while keeping physical applications in mind, we have to find answers to the following three formal questions. First, what is the appropriate geometrical and analytical formalism to account for the local, but non-global perturbations of the volume by deformation of its boundaries? Second, how can we quantify the smallness of those perturbations and relate them to solving the model equations in the unperturbed state of the system? Third, how can we account for time dependency of the perturbations in an operator-theoretic perturbation formalism.

Again from the viewpoint of applications in the sciences, it is desirable to find a geometrical answer to the question of how we may handle weakly curved deformations of the boundaries. Namely, we ask under what conditions is it appropriate to neglect the local curvature properties of the vibrating boundaries with respect to the equilibrium boundaries and simply consider their normal displacement from the unperturbed state?

Strategy

Our strategy consist of three parts. A geometrical one for the geometrical setup, a physical one for finding the appropriate model, and an analytical one for finding a perturbation theory to solve the model equations.

  • Geometry The first part is mostly of geometrical flavor spiced with a bit of topology. It concerns the question of what the appropriate formalism for the geometry of the problem is. This question has not been addressed in the vibrational-acoustics literature so far. We start from considering sub-manifolds of a surrounding Euclidean space as models for the volume under consideration. Specifically, we model the time-dependency of the problem by using smooth -parameter families of such manifolds.

    One family is just the constant family mapping each point in time to he unperturbed volume. The other family is which maps the time to the deforming volume. The reason why we start from this geometrical construction is that the time-evolution of the manifolds, i.e., can be observed in experiments most easily. We then ask where a wave equation for the acoustic pressure lives geometrically. In the textbook literature on partial differential equations, the acoustic wave equation is usually considered to live on a product manifold consisting of a base manifold as a model for the time coordinate and a manifold embedded in which is the living space of the the spatial dependencies of the acoustic pressure, i.e., the solution to the acoustic wave equation.

    In a geometrical language, the living space of the solution is a product manifold which is an example of a fiber bundle. Likewise, we can endow under suitable constraints on the topology of ,namely that it is unaffected by the perturbations, the object with a fiber bundle structure. Intuitively, a fiber bundle is just a manifold which admits locally a product structure of a base manifold, the time axis in our setup and a fiber manifold glued to each point of the time axis.

    Geometrically, we then only need to relate , the stationary fiber bundle to the perturbation bundle . The requirement to be made is that the topology of and and of the fibers and are left invariant by the perturbations aids at relating by means of bundle and manifold diffeomorphisms and globally.

    Using the imbedding space for the bundles, we give using the topological constraints and the therefore global Gauss map for the fibers and the Lorentzian metric originating from restriction of the Lorentzian metric of the imbedding space , the -dimensional Minkowskian space. Combing the metric and the diffeomorphism, we are able to reduce the problem to comparing the reference bundle with time-independent metric to the pull-back bundle with time dependent metric. This will be the starting point for the derivation of the perturbation operator in the model equations.

  • Physics Physically, the theories relevant to our present modeling belong to continuum field theory. On the one hand, the acoustic pressure is quantity of the acoustic limit of fluid dynamics assuming irrotational, isentropic and inviscid fluid flow. We start by a modification of a literature action functional of this theory of fluid dynamics to curved space-time. Using the theory of differential forms, we can derive the Euler equations and by acoustic linearization we recover a scalar wave equation in curved space-time - the acoustic wave equation. The dynamics of the boundaries, i.e., the boundary vibrations, can be well described by the motion of a massive membrane-like structure comparable to bio-membranes which we have in mind as prototypical applications of our theory.

    Our starting point is an action functional including the variation of area of the boundaries and including by an approach inspired by Chern-Weyl theory or, more physically, (Dirac-)Born-Infeld electrodynamics, curvature effects, we arrive at a differential equation which is of second order in time and fourth order in spatial variables. The equation reproduces in suitable limits the bending membrane equation derived before and the conservative flexible membrane equation.

    To include dissipation, we transfer the concept of time-lapsing from general relativity to continuum field theory and modify the derivation of the boundary vibrations equation accordingly. In the end, we discuss how to include boundary and initial conditions as well as source terms to our model equations. A Cauchy-Kovalevskaya argument shows that we can convert boundary conditions in source terms and vice versa. We answer the question of how to model the practically more relevant case of localized boundary vibrations: Localized boundary vibrations are boundary vibrations which are non-zero only on one or more mutually dis-connected sub-manifolds of the boundary of the unperturbed fiber .

  • Analysis The analytical part is the bridge between the geometrical and the physical part. Physically, we are interested in solving the model equations and a convenient tools is perturbation theory as developed for quantum mechanics by Dirac [22, 23], Dyson [24, 25] (see also Reed and Simon [81, Section X.12]), and a decade later later by Magnus [58, 8], whose method has been explained nicely by Blanes et al. [7] and has been refined recently by Fernández [29].

    We will present a detailed derivation of a time-dependent perturbation theory in the spirit of Magnus, which then is applied by using Duhamel’s principle and Banach’s fixed-point theorem in the spirit of Dirac’s perturbation ansatz. The relevant mathematical theory is the theory of semigroups of operators, operator evolution equations, and perturbation theory for linear operators applied to “classical” partial differential equations. As a check of our formalism, we compare our results with the literature. Particularly, for the conceptual basis of the analytical perturbation theory, with the theories of Brillouin [9], Fröhlich [33], and Cabrera [11, 12] as well as Feshbach [30], which partially date back to the thirties of last century.

    From the result-oriented point of view, we compare our approach with the perturbation theory developed by Deng and Li [56] more recently using the seminal work of Beale, which has since then undergone refinement by Casarino et al. [15], Gal et al. [34], once more Casarino et al. [16], and many more who are to remain unnamed here.

    In our perturbation theory, we use a decoupling argument that implicitly assumes the smallness of a certain parameter. Upon formulating our model, we will see that this is the case in a large class of vibrational acoustics models; particularly, those used in bio-acoustics and (classical) vibrational acoustics. The final consistency check will be performed by comparing the detailed results we are going to obtain in the general formalism with those that have been achieved in our previous papers [101, 98, 99]. In doing so, we will verify as to whether the semigroups obtained in this paper agree with the semigroups we found before when specializing to the model of internally coupled ears, ICE for short.

    In order to address the convergence issues of the semigroup, we use the concept of analytic vectors to re-obtain a result on the convergence of the Magnus series comparable to a result proved recently by Batkai et al. [2] from the Nagel group. However, we are not interested in initial value problems but rather in inhomogeneous problems so that we could not use the aforementioned result directly. Furthermore, we modify a result obtained by Fernandez [82, 28] concerning the convergence of the Magnus series for bounded, normal operators. We sketch a proof that for weakly perturbed symmetric operators, the same criterion can be applied.

A small Lie- and operator-algebraic digression is needed during the treatment to derive some tools that we need for the further solution of the problem.

Background literature

Since we do not assume that the reader is familiar with all of the pre-requisites that we use in this paper, we give a physically inspired list of textbooks and articles that we found useful in understanding the underlying mathematics. Typically, we start from a few general references [39, 40, 89] and pave our way (see below) to the more rigorous treatments.

  • Geometry and topology A good start to learn classical differential geometry is [91]. A modern approach including a discussion of differential forms to the extent we need it is found in [63]. The classic reference in Riemannian geometry with emphasis on an analytic approach is found in [51]. A nice presentation of the theory of fiber bundles up to Chern-Weyl theory is given by Baum [3]. The interplay between geometry and physics is surveyed from the perspective of a mathematician by Jost [50] and from the perspective of a physicist by Nakahara [67].

    The topological preliminaries can be found in condensed version, mostly omitting detailed proofs but giving detailed examples in [83] - we think that the reference is appropriate given the limited amount of topology we use. In order to see physicists using differential geometry in a hands-on-manner, we refer the mathematical audience to [72, 68, 76] which combines solution strategies to differential equations on manifolds with high-energy physics. The theory presented in the above references has some formal analogies to our theory and one of the authors has also worked in this fascinating field.

  • Partial and ordinary differential equations The theory of partial differential equations is summarized concisely in [49], an applied approach on solutions in the Green’s functions formalism can be found in the encyclopedic works of Polyanin [77, 79, 78]. Practical and theoretical ordinary differential equations can be found in Polyanin et al. [80, 26] and Emmerich [26] to the extent we need it.

  • Functional analysis and Magnus expansion Functional analysis can be abstract such that we used the applied introductions in [103, 104]. The Magnus series is still absent in the graduate physics-textbook literature so that we refer to the pedagogic introduction [7] and the more intensive survey [8] as well as the original paper [58]. Convergence issues have been addressed elsewhere [17, 55].

  • Operator evolution equations and acoustic boundary conditions Operator evolution equations have been expounded in an introductory manner by Emmerich [26], in a more advanced way by Reed and Simon [81], and in full detail by Engel and Nagel [27]. The foundations of acoustic boundary conditions (ABC without dynamics) has been treated in detail by Beale [6, 4, 5].

  • Quantum mechanics and perturbation theory We simply refer to Dirac’s classic [23, §§44-46] for a physical introduction to perturbation theory. A more modern, and lucid, account of the Dyson series has been given by Zeidler [104]. One may consult the older literature [24, 25] for the original motivation. The theoretical physics up to the point we use it as guideline for the mathematical formalism is presented in a more mathematical manner by Scheck and others [87, 86, 85, 84, 88].

  • Acoustics, hydrodynamics and elasticity We refer to the textbooks by Howe [45, 44, 43, 46] covering the acoustics and hydrodynamics we need. For membrane elasticity, we refer to introductions published elsewhere [96, 73]. For an investigation of acoustic applications of parts of our formalism we like to mention the articles [56, 71, 70].

  • Internally coupled ears and bio-acoustics Bio-acoustics has been surveyed by Fletcher [32] on the basis of linear systems theory combined with a purely harmonic input varying like and impedances as fit parameters. A quick introduction to the ideas underlying the model of internally coupled ears (ICE) is available [94]. The internally coupled ears model has been both presented physically [101, 98, 99] and analyzed mathematically from the point view of perturbation theory [42].

The reader interested in the biology of the ICE model is referred to elsewhere [61, 60, 18, 20, 105, 19, 95].

Technical prolegomena

Technically, we will be as rigorous as possible but without sacrificing the applicability of the theory. The main guideline is that physical insight is more valuable than overly pedantic mathematical rigor. For instance, we do not prove that the Laplace-Beltrami operator is a normal operator, which would be needed for one-hundred-percent completeness but is a rather straightforward exercise in functional analysis applied to partial differential equations.

In general, we do not prove results that other authors of textbooks or research literature have obtained, unless we think the argument provides insightful to the theory developed in this paper. Rather we provide a reference. Furthermore, we will not adopt the dull definition-lemma-proof (non-)example style pervading (pure) mathematics textbooks but conform to the more applied literature that physicists and engineers use instead. Nevertheless, mathematical rigor will not be sacrificed fully.

We are confident that the applied mathematicians will feel comfortable with this style. Because of the length of the present paper, we refer the reader for applications to the first [42] in this series of articles, which focuses on acoustic boundary-conditions dynamics (ABCD).

2 Settings and Perturbation-Bundle Theory

Conventions

We use big Latin indices to denote local orthornormal coordinates stemming from on and , small Greek indices to denote the local orthornormal coordinates on the second component of the fiber bundles, and stemming from and small Latin indices to denote local orthornormal coordinates on and stemming from .

Unperturbed bundle

Let be an oriented trivial smooth fiber bundle over such that is imbedded in the -dimensional flat Minkowskian space with . Let the fiber be an -dimensional Riemannian manifold such that is smooth, compact, retractible and oriented. Let have a smooth, compact, oriented, closed and -connected topological boundary such that is imbedded diffeotopically in and is a smooth -dimensional Riemannian submanifold of and with the properties listed above. We call for short the fiber bundle and call it unperturbed bundle.

Perturbations

We assume at the moment that are known. Thus, also the fiber bundle is known. Since is imbedded in , we can interpret every by means of the imbedding as a point in . We choose arbitrary and and consider the fiber in . For every we take the fiber . We move in by an orientation preserving motion, i.e. an affine mapping with such that with respect to the Lebesgue-Borel integration measure where denotes the Borel--algebra on . It is generated by the topology on induced by the Euclidean norm . This process ensures that for all the embeddings map to such that we maximize the -dimensional volume enclosed in the two bounded manifolds. In particular, we can now start comparing and in by means of the Euclidean norm . We further relate to . By smoothness of -dependence of for , the volume depends smoothly on where we denote by the moved copies of the original . Now, we are able to define where the limit is to be understood in the sense that where we recall denotes the smooth -parameter family of (global) parameterizations of for all w.r.t. the coordinates on . Now we define the notion of a perturbation bundle.

Perturbation bundle

Let be the oriented smooth fiber bundle with total space over such that is imbedded in the -dimensional flat Minkowskian space with . Let the fibers be a family of -dimensional Riemannian manifolds with the metric from the fibers of the unperturbed bundle , parameterized smoothly by in the base space and imbedded diffeotopically in , such that are smooth, compact, retractible and uniformly oriented. Let for all have smooth, compact, oriented and -connected topological boundary for all which are Riemannian submanifolds of and of dimension .

We call for short fiber bundle and call it perturbed bundle if there is a real where such that for and , and for a real , it holds that . We call the pertubation strength.

Visualization

The definition of a perturbation bundle is visualized in Fig. 2.

Figure 2: Visualization of the definition of a perturbation bundle. The perturbed fiber, , depicted as the region bounded by the solid red line, is contained for all in the manifolds visualized by the region bounded by outer dashed black line. Further the perturbed fiber contains for all the manifold , i.e. the region enclosed by the inner dashed black line. are by definition smaller resp. bigger sized copies of which is the region enclosed by the solid black line.

Bundle structure of

Observe that is indeed a fiber bundle over . We introduce on the norm topology such that is the topology generated by all open intervals . The projection map is given by . By diffeotopy of , the imbeddings give rise to a diffeomorphism with non-equality by .

Since each is retractible, smooth and bounded topologically by , we have further a diffeomorphism . Comparing dimensions, we can choose to be a proper imbedding. That is, , where is the unit sphere in . Introduce in order to obtain after localization -dimensional polar coordinates on . We have via a diffeormorphism .

Differentiability of follows from the assumptions on smoothness of and bijectivity follows from bijectivity of the projection map on the first component, differentiability of is a consequence of the inverse function theorem. Even more,we have globally because and we can choose to have the same orientation for all by uniform orientedness. I.e., is a trivial bundle over .

Manifold structure of :

Since is a trivial bundle,it inherits from and the product manifold structure. Since further by assumption and , we can define a metric of Lorentzian signature on by , i.e., by pulling back on the Minkowski metric induced on by restriction of the standard Minkowskian metric on . In components, we have

(4)

where denotes the metric element of . In matrix representation,

(5)

where is the metric on , given by . I.e., is the pullback of the -dimensional Euclidean metric on to . Since the -dimensional spherical coordinates can be equally taken to form a coordinate system of , we can extend the metric from to all of .

Link between and

In this paragraph, we explain Fig. 3. The diffeomorphism between the fiber bundles and the fibers of the respective fibers are depicted. Right column: Because of the topological requirements on the an the , each , is properly diffeomorphic to the closed unit ball in . By means of the global diffeomorphism and , the fibers of the two bundles and are properly diffeormorphic to each other, . Left column: Because and are fiber bundles, they are at least locally diffeomorphic to . Because of topological obstructions, namely the boundary for all , these diffeomorphisms are only local.

By orientedness of the fiber spaces and orientedness of the total spaces of the bundles, a global diffeomorphism, even more bundle biffeomorphisms can be defined: , . These can be composed using bijectivity of the bundle morphism to obtain a bundle morphism . By means of the projection maps and , open subsets of the total spaces can be mapped to open subsets of the base space . Since by assumption, and are imbedded in a Euclidean space and of equal dimension as the total spaces resp. fibers, and thus are equipped with the corresponding relative topologies, the single point can be obtained by means of the intersection of all its -neighborhoods in starting at small enough .

Inverting the projections , the fiber bundles ”evaluated” at the point , and is obtained. The reference bundle is, as a product manifold, a trivial bundle over with fiber , such that can be included in a canonical way in the total space . By means of the projection on the second component, , of the product spaces , the manifolds are obtained directly from the fiber bundles . Fig. 4 contains Fig. 3 and the Minkowski space as the overall imbedding space of the bundles and .

Figure 3: Transformations between and See the main text for explanations.
Figure 4: Overview of the geometric ingredients used in the perturbation theory and the relations between them. See the main text for explanations.

-periodicity

We call a perturbation bundle -periodic if we have and with regularity as subsets of such that the fiber bundle

has a quotient bundle such that , i.e., is the original perturbation bundle. The quotient operation is understood to act on the base space of .

Explanation

Let use define the bundle including the time-boundaries and as follows

using the -periodicity condition with regularity. The bundle has the advantage that when we consider later on a wave equation with boundary and initial condition, we can regard the initial conditions as boundary conditions on and using the isomorphy where . By means of this identification, we identify . By smoothness of , thus of we obtain smoothness of

except possibly at . However, the bundle is at least continuous by the -periodicity condition. By the setting, we further have regularity of the limit . Thus is a differentiable fiber bundle. From the theory of differentiable manifolds and bundles, we know that we can choose for manifolds or bundles even a atlas. Thus, can be turned again in a bundle. By definition, we can identify this with , namely up to a Lebesgue null-set w.r.t. the Lebesgue-Borel integration measure . The workflow is depicted in Fig. 4.

Figure 5: The workflow used to include the initial conditions of a wave equation on . The Roman numbers in blue correspond to the individual steps explained in the main text.


-periodicity and physical relaxation: A short comment on the interpretation of this identification is in order: Physically, the fibers start deforming smoothly at from and approach again at . The identification tells us that the bundle is ready again to go through the entire deformation process. In view of the ICE model [98, 99, 101, 94, 95, 42], this means that after one exposition of the gecko to sound stimulus, the gecko can be exposed to another sound stimulus such that the hearing system of the gecko has forgotten that there was a prior sound stimulus (no echo). The requirement which was needed for the identification means that the deformations relax again, i.e., are damped in some sense. In the ICE model [98, 99, 101, 94, 95, 42] this corresponds to the deformations that the cylinder undergoes by membrane vibration. Since the membranes are damped because the gecko should ultimately stop hearing a certain stimulus from at , the cylinders return at to their equilibrium shapes at , i.e., .

Proper perturbation bundles

We call a perturbation bundle proper if for all and in the imbedding space and for all . Here denotes the Levi-Civita connection on induced by the metric . It is now time for a few motivating explanations.

If , . The definition formalizes the intuition that given local coordinates in and whereas with a pointwisely non-zero function depending on the local coordinates on and, in general, the time-parameter . Indeed, , whereas spans . Thus, we have the decomposition of . We visualize the situation as modeled in the following piece of graphics, Fig. 5. Notice that and are nothing but the Gaussian mapping from -dimensional (imbedded), oriented and closed submanifolds to the oriented unit sphere .

Figure 6: Visualization of the definition of a proper perturbation bundle. The vector is perpendicular to for all and in the target space of the imbeedings for .

Fibratable perturbation bundles

We call a proper perturbation bundle fibratable, if the diffeomorphism induce diffeomorphisms and such that is smooth in . The symbols denote the unit normal vectors to and .

By means of the Gauss maps and every fiber , , allows the diffeomorphisms in the above equation by pulling back by the Gauss maps. Thus, every proper perturbation bundle is fibratable. That is, can be fibrated. Let is quickly make a digression to what fibratable means in the present context. For the unit ball the -sphere gives rise to a fibration of the following way. Let us denote an element of by and define by

(6)

where is the outward unit normal vector to as a -submanifold of co-dimension . In order to ensure bijectivity, we had to exclude the center of , i.e., . However, is a Lebesgue-Borel null-set w.r.t. the Lebesgue-Borel measure restricted to such that this does not hinder our further progress. Now, induces an analogous fibration on if

(7)
(8)

In the previous equation, a point on the boundary is denoted and a point in by . Analogously, induces a fibration on is required by the definition if

(9)
(10)

where and denotes a generic point in . Denoting symbolically and similarly , we can write

(11)

The geometric intuition stored in the definitions of is illustrated in the subsequent figure (Fig. 6).

Figure 7: The unperturbed domain and its boundary together with the smaller sized for a fixed . The points are elements of . denotes for the inward unit normal vector at on (blue vector). Staring from , we move by an amount of along inwards (green vector). The result is depicted by the violet vector. The necessity to take instead of simply for the amount we move inwards is depicted by the golden line. has in general a non-zero tangential component, where tangential means tangential to . For a general , the position vector points to but, regarded as vector field, is neither element of nor but only where the orthogonal complement of the sub vector space is to be taken in .

Physical perturbation bundles

We call a proper perturbation bundle physical if there is a (dimensional) constant satisfying and such that in the metric continued to , conservative if the above inequality is an equality and dissipative if the above inequality is strict.

Explanation: The property of a perturbation bundle to be physical states that the internal energy density of the perturbations is non-increasing. We consider the boundary to be a membrane. In the physical picture, membranes can be thought of as a continuum of harmonic oscillators coupled to each other. This gives rise to the potential energy density where is a physical constant, the membrane tension. The vibration of the membranes on the other hand gives rise to a kinetic energy density noting that if is brought into diagonal form. denotes a physical constant, the surface mass density of the membrane. The property “conservative” states that each oscillator can convert the entire potential energy into kinetic energy. The property “dissipative” expresses the physical reality that there are heat losses due to internal friction effects between the individual oscillators and external friction due to the membrane interacting with its environment which hinder the conversion of all potential energy into kinetic energy on a local level, i.e., for each oscillator. I.e., . Equating we recover the definition. The condition now turns into . For the values given in [98] it can be fulfilled by choosing a suitable .

The condition expresses as the admission of the boundary waves to be transmitted without information loss through the cavity. This requires that the speed of propagation of the membranes vibration, , is much smaller than the speed of propagation of the wave inside the gecko’s interaural cavity, . Otherwise, the membranes would vibrate so fast that not all cavity eigenmodes within the audible frequency range of the gecko have the chance to transmit the information of membrane vibration. Physiologically, it is only the lowest eigenmodes that the gecko can perceive by its hearing system.

Convention: Henceforth, we let denote a proper and physical perturbation bundle with fibratable fibers around .

Connection to fluid mechanics:
The original ICE model which serves as a physical role model for our mathematical structures has been formulated in terms of physical quantities, namely the acoustic pressure and the membrane displacements . It is natural to ask how the geometric approach relates to the acoustic quantities. At first, we take the diffeomorphism and perform a rewriting using bijectivity and smoothness of diffeomorphisms as well as the convenience of the definition of a perturbation bundle that the relevant fibers and are imbeded submanifolds of . We let and consider the subsequent equation as a vectorial equation in :

(12)

We use this equation to make contact to fluid dynamics in a twofold way. Firstly, we isolate the second term in the equation and restrict it to the boundary . We recall the previously introduced notation and . Further,we use that are proper in order to justify for the notation and . Using regularity of the perturbation bundle, we have for all ,