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We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat M\"obius strip are determined by their lens data. We also see by example that the flat M\"obius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid., Comment: 12 pages, 2 figures

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}{\pi}R^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $\mu_2(B(p,R)) \leq 2(\frac{\pi}{2 R})^2$ on the first nonzero Neumann eigenvalue $\mu_2$ of the Laplacian in terms only of the radius. This has also been conjectured for $R$ up to half the injectivity radius., Comment: 4 pages

This paper considers Riemannian metrics on two dimensional disks where all geodesics are minimizing. A sharp reverse isoperimetric inequality is proven. This in turn yields near optimal bounds for the area of disks as well as near optimal upper bounds on the first nonzero Neumann eigenvalue of the Laplacian in terms only of the radius.

We consider compact Riemannian manifolds ( M , ∂ M , g ) (M,\partial M,g) with boundary ∂ M \partial M and metric g g on which a finite group Γ \Gamma acts freely. We determine the extent to which certain rigidity properties of ( M , ∂ M , g ) (M,\partial M,g) descend to the quotient ( M / Γ , ∂ / Γ , g ) (M/\Gamma ,\partial /\Gamma ,g) . In particular, we show by example that if ( M , ∂ M , g ) (M,\partial M,g) is boundary rigid, then ( M / Γ , ∂ / Γ , g ) (M/\Gamma ,\partial /\Gamma ,g) need not be. On the other hand, lens rigidity of ( M , ∂ M , g ) (M,\partial M,g) does pass to the quotient.

A fully reversed (FR) sequence of rotations is defined as a series of rotations of a free rigid body about its body-fixed axes such that the rotation about each axis is fully reversed at the end of the sequence. Due to the non-commutative property of finite rigid body rotations, an FR sequence can effect non-zero changes in orientation of the rigid body even though the net rotation about each axis is zero. The FR sequences are useful for attitude maneuvers of miniature spacecraft that use elastic deformation-based microactuators and for other airborne or neutrally buoyant underwater vehicles where actuations effecting orientation change are restricted. This paper considers the kinematics of six-rotation FR sequences and discusses if each of them is capable of producing any desired change in orientation. It is concluded that out of a total of 24 six-rotation sequences, 12 are capable of this but not the remaining 12. The results are proved using mathematical formalism and are also interpreted using numerical computations and graphical visualization.

We show that the metric of non-positively curved graph manifolds is determined by its geodesic flow. More precisely, we show that if the geodesic flows of two non-positively curved graph manifolds are C 0 conjugate, then the spaces are isometric.

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points., Comment: The revision improves the main result and simplifies the proof

Here we present a method for the coupled design of four freeform reflective surfaces that will control a bundle of rays. By this, we mean that given an input bundle of rays, we can construct an optical system that will map it to a given output bundle, where a ray-to-ray correspondence is realized as per the prescribed data. The method makes use of the Cartan–Kahler theorem of exterior differential systems. Sample imaging applications are given.

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold ( M , g ) (M,g) with boundary ∂ M \partial M whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics g ′ g’ on M M there is a C 3 , α C^{3,\alpha } -neighborhood of g g such that g g is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance — as measured in M M ). More precisely, given any metric g ′ g’ in this neighborhood with the same boundary distance function there is diffeomorphism φ \varphi which is the identity on ∂ M \partial M such that g ′ = φ ∗ g g’=\varphi ^{*}g . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.

We construct a pair of finite piecewise Euclidean 2-complexes with nonpositive curvature which are homeomorphic but whose universal covers have nonhomeomorphic ideal boundaries, settling a question of Gromov.

In [1] V. Guillemin and D. Kazhdan introduced the following definition of spectral rigidity of a Riemannian manifold. Let (M, g) be a compact boundaryless Riemannian manifold. A family g of Riemannian metrics on M smoothly depending on the parameter τ ∈ [−2, 2] is called the deformation of the metric g if g = g. A deformation is called trivial if there exists a one-parameter family of diffeomorphisms φ : M → M such that φ = Id, and g = (φ )∗g0. Given a deformation g (−2 ≤ τ ≤ 2), let ∆ : C∞(M) → C∞(M) be the Laplace-Beltrami operator corresponding to the metric g . The deformation is called isospectral if, for all −2 ≤ τ ≤ 2 spectra of the operators ∆ and ∆ coincide (counting multiplicities). A Riemannian manifold (M, g) is called spectrally rigid if it does not admit non-trivial isospectral deformations. The main result of the present article is the following

In this paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, Section I.5]. The main theorem is that if $X$ is a complete, simply connected Riemannian manifold without conjugate points, and $M=X\times R$ is given the Riemannian product metric $g$, then any metric without conjugate points on $M$ which agrees with $g$ outside a compact set is isometric to $g$.

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

Let M and N be compact Riemannian manifolds with sectional curvature K ⩽ 0 such that M has dimension ⩾ 3 and rank ⩾ 2. If there exists a C 0 conjugacy F between the geodesic flows of the unit tangent bundles of M and N , then there exists an isometry G : M → N that induces the same isomorphism as F between the fundamental groups of M and N .

In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\gamma_V(t)$ again lies in the boundary. We show that any other Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to $\partial(D^n\times S^1)$ and with the same scattering data must be isometric to $D^n\times S^1$. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit tangent bundle., Comment: 12 pages, 1 figure

We construct a counterexample to a conjectured inequality L, 10 pages; to appear in Geometric and Functional Analysis

The notion of optimization is inherent in the design of a sequence of amino acid monomer types in a long heteropolymer chain of a protein that should fold to a desired conformation. Building upon our previous work wherein continuous parametrization and deterministic optimization approach were introduced for protein sequence design, in this paper we present an alternative formulation that leads to a quadratic programming problem in the first stage of a two-stage design procedure. The new quadratic formulation, which uses the linear interpolation of the states of the monomers in Stage I could be solved to identify the globally optimal sequence(s). Furthermore, the global minimum solution of the quadratic programming problem gives a lower bound on the energy for a given conformation in the sequence space. In practice, even a local optimization algorithm often gives sequences with global minimum, as demonstrated in the examples considered in this paper. The solutions of the first stage are then used to provide an appropriate initial guess for the second stage, where a rescaled Gaussian probability distribution function-based interpolation is used to refine the states to their original discrete states. The performance of this method is demonstrated with HP (hydrophobic and polar) lattice models of proteins. The results of this method are compared with the results of exhaustive enumeration as well as our earlier method that uses a graph-spectral method in Stage I. The computational efficiency of the new method is also demonstrated by designing HP models of real proteins. The method outlined in this paper is applicable to very large chains and can be extended to the case of multiple monomer types.

We consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas, with equality implying isometry.

We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp isoperimetric inequality for surfaces with boundary such that every pair of geodesics have at most one interior intersection point., Latex, 4 pages

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X,g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T^b, which are area-decreasing (on b-dimensional areas), with respect to suitable norms. These norms are the stable norm of g, the conformally invariant norm, as well as other L^p-norms. The case of equality is characterized in terms of the criticality of the lattice of deck transformations of T^b, while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, in the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover., 22 pages. Transactions Amer. Math. Soc., to appear

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics., 21 pages, 3 figures, to appear in Geometric and Functional Analysis (GAFA)

The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperimetric inequalities etc.). We will restrict our attention to those results that have a direct connection to the boundary rigidity problem (see Section 2) or the conjugacy rigidity problem (see Section 4). Even with that restriction the connections are numerous and the author was forced to select the topics covered here in accordance with rather subjective criteria. A few of the topics not covered are mentioned in Section 11 but there are others.

Foreword * Preface * On the construction of isospectral manifolds, Werner Ballman * Statistical stability and time-reversal imaging in random media, James G. Berryman, Liliana Borcea, George C. Papanicolaou, and Chrysoul Tsogka * A review of selected works on crack identification, Kurt Bryan and Michael S. Vogelius * Rigidity theorems in Riemannian geometry, Christopher B. Croke * The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, R. Gulliver, I. Lasiecka, W. Littman, and R. Triggiani * Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds, A. Katchalov, Y. Kurylev, and M. Lassas * Ray transform and some rigidity problems for Riemannian metrics, Vladimir Sharafutdinov * Unique continuation problems for partial differential equations, Daniel Tataru * Remarks on Fourier integral operators , Michael Taylor * The Cauchy data and the scattering relation, Gunther Uhlmann * Inverse resonance problem for Z2-symmetric analytic obstacles in the plane, Steve Zelditch * List of workshop participants

In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary, allows the survey to be reasonably short. Although, even in this limited case the authors have left out many interesting results., Comment: 34 pages. To appear in Surveys in Differential Geometry

We consider discrete cocompact isometric actions $ G \curvearrowright^{\rho} X $ where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action $ \rho $ which determine, and are determined by, the equivariant homeomorphism type of the action $G \curvearrowright^{\partial_\infty \rho}\,\partial_\infty X X $ of G on the ideal boundary of X. Moreover, if $ G \curvearrowright^{\rho}i X_i $ are two actions with the same geometric data and $ \Phi : X_1 \to X_2 $ is a G-equivariant quasi-isometry, then for every geodesic ray $ \gamma_1 : [0, \infty) \to X_1 $ there is a geodesic ray $ \gamma_2 : [0, \infty) \to X_2 $ (unique up to equivalence) so that $ {\rm lim}_{t \to \infty} {1 \over t}\,d_{X_2}(\Phi \circ \gamma_1(t), \gamma_2([0, \infty))) = 0 $ . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].

The set of rigid body motions forms a Lie group called SE(3), the special Euclidean group in three dimensions. In this paper we investigate possible choices of Riemannian metrics and affine connections on this manifold. In the first part of the paper we derive the semi-Riemannian metrics whose geodesics are screw motions and show that the only metrics are indefinite but non degenerate, and they are unique up to a choice of two scaling constants. In the second part of the paper we investigate affine connections which through the covariant derivative give the correct expression for the acceleration of a rigid body. We prove that there is a unique symmetric connection which achieves this. Further, we show that there is a family of metrics that are compatible with such connection. A metric in this family must be a product of the bi-invariant metric on the group of rotations and a positive definite constant metric on the group of translations.

The celebrated Cheeger-Gromoll splitting theorem (see [ChGr], and also [EsHe, Wu]) implies that a connected, complete Riemannian manifold having nonnegative Ricci curvature and two ends is isometric to a Riemannian product, where one factor is a line. There is a version of this theorem for compact, connected Riemannian manifolds M with two smooth boundary components. It states that if M has nonnegative Ricci curvature and two boundary components, both of which have nonnegative mean curvature with respect to their inward normals, then M is isometric to the Riemannian product of N1 with a closed interval (in particular N1 is isometric to N2). In this paper we will prove an analog of this result for manifolds with Ricci curvature ≥ −(n − 1). Since our argument parallels the proof of the splitting theorem for manifolds with nonnegative Ricci curvature, we present the proofs simultaneously by letting δ be either zero or one in what follows.

In this note we improve a previous result to show that euclidean nspace (n > 3) is the only simply connected manifold without conjugate points which is flat outside a compact set. Our goal is to prove the following: Theorem. If M is a complete, simply connected Riemannian manifold of dimension n > 3 which has no conjugate points and is flat outside a compact set, then M is isometric to IRn. This is an improvement of Theorem A of [C], where we prove the same theorem under the assumption that the complement of a compact set is isometric to the complement of a compact set in IRn . The proof here consists of reducing the above theorem to Theorem A of [C]. The theorem is clearly false for n = 2. However, as had been shown earlier in [G-G], Theorem A of [C] does hold in this case as well. Proof. Pick x E M, and choose R so large that M B(x, R/2) is flat, where B(x, R) represents the ball of radius R about x. Let S(x, R) be the boundary of B(x, R). Since M has no conjugate points and is simply connected, M is diffeomorphic to RIn, B(x, R) is diffeomorphic to an n-ball, and S(x, R) is diffeomorphic to an n 1 sphere. We claim that the second fundamental form of S(x, R) is definite. Let y E S(x, R) and y be the geodesic with y(O) = x, and y(R) = y. Let e be an eigenvalue of the second fundamental form of S(x, R) at y, and let X E T7S(x, R) be a corresponding eigenvector. If T(S) E S(x, R) is a curve with T'(0) = X, then the variation of geodesics H(s, t) = ExpT(S) (t-R)N(T(s)), where N(T(s)) is the outward normal to S(x, R) at T(S), has as variation field the Jacobi field J(t) with J(R) = X and J'(R) = eX. Further, since H(s, 0) = x, J(O) = 0. Since there are no conjugate points, J cannot vanish for t :# 0 and, by [E-O], must get arbitrarily large as t -+ oo. For t > R, M Received by the editors September 22, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20: Secondary 53C25. Research supported by NSF grant DMS87-22998 and M.S.R.I. ? 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page

Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center pOint?XX Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric on R4 (extending to higher dimensions) of positive RiCCi curvature, whose sectional curvatures decay to 0, and such that the volume of balls goes uniformly to 0 as the center goes to infinity. 0. Introduction. The following question came up in discussions of the first author with Peter Li: Under what curvature assumptions on a complete open manifold M can one find a constant c(M, R) such that the volume of balls B(x, R) can be bounded from below independent of the midpoint x, (0.1) vol(B(x, R)) > c(M, R)? Note that in the presence of a lower bound on the Ricci curvature one needs to prove the above inequality only for some R since the Bishop-Gromov volume comparison theorem [Gr, p. 124] then gives a bound c(M,r) for r c(n) * rn. In dimension 2, Theorem A below is a satisfyingpositive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples. THEOREM A. If M2 is complete with curvature K > O then there exist c(M) such that for radii R c(M) R2. In higher dimensions however we have EXAMPLE 1. There are convex hypersurfaces Md in Rd+1 such that for d > 3 no lower bounds as in (0.1) exist for Md. In dimensions > 2 we therefore assume an upper curvature bound. But then one gets uniform control even on the injectivity radius. The following theorem has been rediscovered several times. Received by the editors August 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20; Secondary 53A07. Research supported by NSF Grant MCS 7g01780 and the Sloan foundation. Both authors were supported by invitations to the I.H.E.S. and the M.S.R.I. (r)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

In 1958 Lichnerowicz [7] showed that for a compact n-dimensional riemannian manifold M, whose Ricci curvature is bounded below by n 1 , the first non-zero eigenvalue, 21, of the laplacian satisfies 2 t>n . If, in fact, 21 =n Obata proved that M must be isometric to the standard sphere. A natural question is: Do there exist constants C(n)> 1, depending only on n such that if M is as above and C(n). n> 21 > n then M must be diffeomorphic to a sphere. Here, by combining the works of Gromov [3], Berard and Meyer [1], and Grove and Shiohama [4], we show

In this paper we consider metrics of the form d s 2 = d r 2 + f 2 ( r , θ ) d θ 2 d{s^2} = d{r^2} + {f^2}(r,\theta )d{\theta ^2} on a ball of dimension n ⩾ 3 n \geqslant 3 . We show that if the diameters (geodesics through the origin) minimize length then the volume of the ball is larger than the volume of the hemisphere of the corresponding round sphere. This relates to a conjecture first considered by Marcel Berger. We also give examples in all dimensions of radially symmetric metrics on balls of radius π / 2 \pi /2 having arbitrarily small volume and yet having no pair of points conjugate along a diameter.

We generalize work of J.P. Otal and C. Croke on the marked length spectrum of surfaces to the case where one of the metrics is of nonpositive curvature and the other one has no conjugate points. If M is a manifold and g1, g2 are two Riemannian metrics, we say that they have the same marked length spectrum if in each homotopy class of closed curves in M the infimum of g1-lengths of curves and the infimum of g2-lengths of curves are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. Of course, this cannot hold for arbitrary metrics (for example if M is simply connected). This problem was stated as a conjecture in [BK] in the case where M is a closed surface and g1 and g2 are of negative curvature. This conjecture was solved by J.P. Otal [To] and independently by C. Croke [Cr]–see also [Fa]. Previous work on the problem was done by Guillemin and Kazhdan [GK]. In this work, using Otal’s approach, we improve some of these results by proving the following theorem: Theorem A. Let M be a closed surface and let g1, g2 be Riemannian metrics on M , with g1 of nonpositive curvature and g2 without conjugate points. If g1 and g2 have the same marked length-spectrum then they are isometric by an isometry homotopic to the identity. We will also prove the following fact, which reduces the length spectrum and curvature condition to the assumption that the Morse correspondence preserves angles—see §1 for the definition of the Morse correspondence. Theorem B. Let M be a closed surface of genus ≥ 2, and let g1, g2 be Riemannian metrics without conjugate points on M . If g1 and g2 have the same marked lengthspectrum and the Morse correspondence preserves angles then they are isometric by an isometry homotopic to the identity.

On demontre que pour la metrique standard sur RP n l'application identite est minimisante de l'energie dans sa classe d'homotopie