本书是一部英文版的数学专著，中文书名或可译为《各向异性黎曼多面体的反问题：分段光滑的各向异性黎曼多面体反边界谱问题：唯一性》。
有些物理系统的参数，例如密度、电导率，也许并不知道也不能被直接测量。为了得到这些参数的数值，以至系统可以被尽可能完全的理解，我们解决这个反问题或者知果求因。反问题最早来自地球物理（分析地球的内部或者石油分布）、医学成像（MRI，超声）、遥感、海洋声学层析成像、非破坏试验和天文学。为了与应用紧密联系，我们将物理系统模型化为具有分段光滑系数的在各向异性的黎曼多面体上的偏微分方程系统，这样的多面体是有许多不同的材料“粘”在一起所构成的。反问题于是变成决定多面体的结构、度规和给出部分在可知点（如地球的表面）特解信息偏微分方程系统的系数。在这本书里解决的唯一性问题和介绍的技术在数学中有着巨大的应用并且会吸引任何对现代交叉学科研究感兴趣的人。

1 Introduction
1.1 Inverse problems
1.2 Background
1.3 Outline of this work
2 Geometric Description
2.1 Basic notations
2.1.1 Admissible polyhedron
2.2 Admissible Riemannian polyhedron
2.2.1 Metric structure
2.2.2 Coordinates
2.3 Distance structure on (M4,g)
2.3.1 Intrinsic distance dx
2.4 Laplace-Beltrami operator
2.5 Spectral problem
3 Gaussian Beams near the Interface
3.1 Gaussian beams (" quasiphotons" )
3.1.1 Solution form
3.1.2 Main results
3.1.3 Formal series
3.2 Phase functions
3.2.1 Main equations
3.2.2 Required preparations
3.2.3 Impulses pref ptr construction
3.2.4 Qua~tratic forms
3.2.5 Phase functions ON (t,q,σ), Ntr(t,q,σ)
3.3 Reflection and transmission laws
3.4 Amplitudes
3.4.1 Amplitude values on the interface
3.4.2 Amplitude equations
3.4.3 Initial values
3.5 Exact and Approximate Solutions Estimates
3.5.1 Convergence
3.6 Conclusion
4 Reconstruction of a Smooth Manifold
4.1 BSD on the entire boundary
4.1.1 Formulation of the inverse problem for a smooth Riemannian manifold
4.1.2 Reconstruction of the Fourier coefficients of thewaves
4.1.3 Domains of influence. Tataru's theorems. Wave basis
4.1.4 On the role of Ganssian beams and boundary distance functions
4.2 IP with data given on a part of the boundary
4.2.1 First submanifold reconstruction
4.2.2 Recalculation of the boandary spectral data of (△p,D)
4.2.3 Reconstruction of M
4.2.4 Iterating procedure; Mm = Mint
5 Uniqueness Problem for the Polyhedron
5.1 Formulation of the uniqueness problem
5.2 The Holmgren-John uniqueness theorem
5.3 Uniqueness inverse problem
5.4 Meeting the interface
5.5 Crossing the interface
5.6 Polyhedra isometry
6 Conclusions and Outlook
References
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