Julià Cufí, Eduardo Gallego, Agustí Reventós
We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke.
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A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
A general method for analytic inversion in integral geometry is proposed. All classical and some new reconstruction formulas of Radon-John type are obtained by this method. No harmonic analysis and PDE is used.
The purpose of this paper is to extend the mean curvature comparison and volume comparison estimates by Petersen, Sprouse, and Wei to integral generalized quasi-Einstein tensor. Moreover, we use our comparison results to get global diameter estimate.
This is a corrected version of my paper "Application of integral geometry to minimal surfaces" appeared in International J. Math. vol. 4 Nr. 1 (1993), 89-111. The correction concerns Proposition 3.5. We discuss this correction in Appendix to the original version of my published paper by reproducing our correspondence with Professor Tasaki.
We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.
We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-Émery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.
In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results.
A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.
In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation on a Riemannian manifold and then the uniqueness theorem is proved in semi-geodesic coordinates by using the tools of Fourier analysis.
This is a revised version of the notes from the week-long course I gave at the Centre de Recerca Matematica, Barcelona, in September of 2010. The aim is to give a working overview of recent methods and results in "Blaschkean integral geometry" (i.e. the subject revolving around the kinematic formulas of Blaschke) in the wake of the revolutionary new methods introduced over the past ten years or so by S. Alesker. The most striking new feature is the extent to which the subject has been invaded by algebra.
In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this …
The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a polynomial algebra. This is a short way to encode all local kinematic formulas. We then characterize the invariant valuations on complex space forms leaving the space of invariant angular curvature measures fixed.
A general method for analytic inversion of geometric integral transforms is proposed
We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of characteristics of the equation $\mathcal{E}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal{E}$. After specifying the results to general Monge-Ampère equations (MAEs), we focus our attentio…
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the algebra associated to this co-product is described explicitly in terms of generators and relations. As a consequence, a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we discuss these manifolds in the later sections of the paper.
We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this appr…
We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.
In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.
The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of SU(2)- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line is stated and proved.
We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmann manifold. Some applications are given.
We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.
The axiom of θ-holomorphic 2-planes is introduced. It is proved, that if an almost Hermitian manifold satisfies this axiom for a fixed θ, 0< θ< π/2, then it is a real space form.
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.
We prove that the product of equators $S^{1} \times S^{1}$ in $S^{2} \times S^{2}$ is globally volume minimizing under Hamiltonian deformations.
We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.
We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group.
We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced to an extremal problem for the harmonic extension of functions and the related nonlinear integral equation is discussed.
The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.
We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation of $w$ implies that locally supported solutions of the steady Euler equation in $\mathbb R^3$ are zero.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.
We construct and study differential and integral calculus on the space of states of a C*-algebra by equipping it with a formal smooth structure. To achieve this goal we first concentrate on the space of nonpure states of a commutative C*-algebra as a guideline for the noncommutative case. In particular, we prove Stokes' theorem over the both commutative and noncommutative smooth Wasserstein space.
We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.
We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedded hypersurface of constant mean curvature must be a geodesic sphere, provided the hypersurface is Hopf. A similar result is established on the complex projective space.
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been investigated intensively for 1<p<\infty, but not for p=1. We show that integral geometry for the 1-norm bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p. We prove a Hadwiger-type theorem for R^n with the 1-norm, and analogues of the classical formulas of Steiner, Crofton and Kubo…
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges …
String geometry theory is one of the candidates of the non-perturbative formulation of string theory. In this paper, from the closed bosonic sector of string geometry theory, we derive path integrals of all order perturbative strings on all the string backgrounds, $G_{μν}(x)$, $B_{μν}(x)$, and $Φ(x)$, by considering fluctuations around the string background configurations, which are parametrized by the string backgrounds.
We study the geodesic X-ray transform $I_Γ$ of tensor fields on a compact Riemannian manifold $M$ with non-necessarily convex boundary and with possible conjugate points. We assume that $I_Γ$ is known for geodesics belonging to an open set $Γ$ with endpoint on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any $(x,ξ)\in T^*M$, there is a geodesic without conjugate points in $Γ$ through $x$ normal to $ξ$.
Path-integrals of perturbative superstrings on curved backgrounds from string geometry theory
本文从弦几何理论出发，通过考虑由弦背景参数化的背景构型周围的涨落，导出了所有弦背景上微扰超弦的路径积分，为弦理论的非微扰表述提供了候选方案。
In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.
This is the introductory part of my PhD thesis on inverse problems arising in medical and seismic imaging. The topics include X-ray tomography of scalar and vector fields with partial data, higher order fractional Calderón problems, travel time tomography on Riemannian and Finsler manifolds, and unique continuation of fractional Laplacians. The thesis additionally includes these articles: arXiv:1909.05585, arXiv:2001.06210, arXiv:2006.05790, arXiv:2008.10227, arXiv:2009.01043, arXiv:2010.11484, arXiv:2103.14385.
Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has zero measure.
The u-plane integral is the contribution of the Coulomb branch to correlation functions of N=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU(2), for an arbitrary four-manifold with (b1,b2+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg-Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined …
The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $\mathrm G_2$ and $\mathrm{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key t…
Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifo…
For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $\mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $\mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral an…
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