Comparison theorems and hypersurfaces

Bergman kernels related to Hermitian line bundles over compact complex manifolds by B. Berndtsson A gentle introduction to points of finite type on real hypersurfaces by J. P. D'Angelo The moduli space of 2-step nilpotent Lie algebras of type $(p,q)$ by P. Eberlein The homotopy principle in complex analysis: A survey by F. Forstneric Finiteness theorems in riemannian geometry by K. Grove Generalized Toponogov's theorem for manifolds with radial curvature bounded below by Y. Itokawa, Y. Machigashira, and K. Shiohama The global isometric embedding problem by H. Jacobowitz The Bergman metric invariants and their boundary behavior by K.-T. Kim and S. G. Krantz Natural connections in almost complex manifolds by S. Kobayashi The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type by S. Kumar, B. Leeb, and J. Millson Subelliptic estimates and scaling in the $\bar\partial$-Neumann problem by J. D. McNeal Negativity of curvature on spaces parametrizing Hodge decompositions of reduced first cohomology groups by N. Mok On the extension of $L^2$ holomorphic functions VI--A limiting case by T. Ohsawa Variations on a theme of Synge by P. Petersen Mappings between real submanifolds in complex space by L. P. Rothschild Harmonic analysis on toric varieties by B. Shiffman, T. Tate, and S. Zelditch On complex manifolds with noncompact automorphism groups by B. Wong Kahler manifolds with slightly positive bisectional curvature by H.-H. Wu and F. Zheng.

The main result can be stated roughly as follows: Let M M be an Alexandrov space, Ω ⊂ M \Omega \subset M an open domain and f : Ω → R f:\Omega \to \mathbb {R} a harmonic function. Then f f is Lipschitz on any compact subset of Ω \Omega . Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.

Motivated by some issues which enter into the Gauss-Bonnet-Chern theorem in Finsler geometry, this paper studies the unit tangent sphere (or indicatrix) Ix M at each point x of a Pinsler manifold M. We demonstrate that the volume of ImM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x. This contrasts sharply with the situation in Riemannian geometry. We also express the derivative of such volume function in terms of the second curvature tensor of the Chern connection. In particular, we find that this function is constant on Landsberg spaces (though that constant need not be equal to the value taken by Riemannian manifolds).

Abstract. Let G be a compact connected semisimple Lie group, g theLie algebra of G, g the canonical metric (the biinvariant Riemannianmetric which is induced from the Killing form of g), and r be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection r in the tangent bundle TG over (G;g) is a Yang-Millsconnection. 1. IntroductionThe problem of nding metrics and connections which are critical pointsof some functional plays an important role in global analysis and Riemann-ian geometry. A Yang-Mills connection is a critical point of the Yang-MillsfunctionalYM(D) =12Z M kR D k 2 v g (1:1)on the space C E of all connections in a smooth vector bundle Eover a closed(compact and connected) Riemannian manifold (M;g), where R D is the cur-vature of D2C E . Equivalently, Dis a Yang-Mills connection if it satis es theYang-Mills equation (cf. [1, 5, 6]) D R D = 0; (1:2)(the Euler-Lagrange equations of the variational principle associated with (1.1)).If Dis a connection in a vector bundle Ewith bundle metric hover a Riemann-ian manifold (M;g), then the connection D given byh(D

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

A sufficiently small neighborhood of a point x in a Riemannian manifold M (and thus, metric space) can be approximated with a neighborhood of the origin in the tangent space (Euclidean) TxM. This fact suggested the idea that usual numerical methods for optimization on Euclidean spaces would be also sufficient as numerical methods on Riemannian manifolds. Our investigations, started in 1976, pointed out the failure of this point of view and the necessity of finding some algorithms which are adequate to the Riemannian structure of the manifold and independent of the choice of coordinate systems. As a matter of fact, the Euclidean conjugate direction method is nothing else than a descent method on a particular Riemannian space (ℝn and a metric with constant components, i.e. , an Euclidean space) . One should also have in mind the approximations of the extrema of energies of the vector fields .

For a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by‐product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

In this paper, for complete Riemannian manifolds with radial Ricci or sectional curvature bounded from below or above, respectively, with respect to some point, we prove several volume comparison theorems, which can be seen as extensions of already existing results. In fact, under this radial curvature assumption, the model space is the spherically symmetric manifold, which is also called the generalized space form, determined by the bound of the radial curvature, and moreover, volume comparisons are made between annulus or geodesic balls on the original manifold and those on the model space.

A cross-session motor imagery classification method based on Riemannian geometry and deep domain adaptation

This chapter introduces fundamental concepts from Riemannian geometry, including the Laplacian and harmonic functions on a Riemannian manifold. The next two sections discuss weak conformality and the dual notion of horizontal weak conformality (or ‘semiconformality’) needed for the description of harmonic morphisms in Chapter 4. The fibres of a horizontally conformal submersion form a conformal foliation, which are studied in the final section.

The sphere theorem in differential geometry has a long history, dating back to a paper by H.E. Rauch in 1951. In that paper [64], Rauch posed the question of whether a compact, simply connected Riemannian manifold M whose sectional curvatures lie in the interval (1, 4] is necessarily homeomorphic to the sphere. Around 1960, M. Berger and W. Klingenberg gave an affirmative answer to this question:

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In this article we study the Jacobi equation associated with the geodesics in a pseudo-hermitian manifold wish vanishing Webster torsion. We develop integral geometric formula generalizing the well known Santalo formula in Riemannian geometry. As applications we obtain volume comparison results under suitable curvature assumptions as well as isoperimetric inequalities for domains in such manifolds.

Euclidean geometry can be understood as a theory that derives from the construction of its fundamental concepts and whose theorems are true on the basis of its evident clarity [Evidenz]. The theorems of Euclidean geometry are therefore strictly valid for all objects of experience. On the other hand, the general theory of relativity shows that for the proportions of empirical objects it is not the Euclidean, but the Riemannian geometry which is valid.The question therefore arises how these two apparently contradictory assertions can be reconciled. To answer this question, the basis for the evident clarity of Euclidean geometry as well as the empirical-physical arguments for the theory of relativity must be considered.KeywordsMass PointRiemannian GeometryEuclidean GeometryRiemannian SpaceEmpirical RealizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let A be an abstract stratification.Assume that every stratum X is a riemannian manifold.We first give some conditions, under which it is possible to endow A with a suitable metric extending a well known technique of riemannian manifolds.Then stronger conditions are introduced in order that the metric space A becomes a G-space.An abstract stratification with all the above conditions is called an abstract riemannian stratification.Whitney stratifications are, in a natural way, riemannian.Introduction.An abstract stratification is the disjoint union of a locally finite set of smooth manifolds, the strata, glued together according to certain rules (seeThe basic tools of differential calculus, smooth functions, vector fields and differential forms, can be also defined for an abstract stratification by means of "controlled" collections of "tools".Classical results can be extended to abstract stratifications (e.g. de Rham's theorems, see [15], [8]) in this way.On the other hand a controlled approach seems to be useless when we take into account a riemannian structure: control conditions on the metric of the strata are too strict (bundle-like metrics, see [11], are not controlled in general).The purpose of this paper is to obtain weaker conditions in order that riemannian metrics of the strata glue together in an appropriate manner.In §1 sufficient conditions are given for extending the fundamental theorem of the riemannian geometry; in §2 a definition of compatibility among the riemannian structures of the strata is given and connections with the structure of G-space are studied.It is shown that the axioms of finite compactness, convexity, local prolongability are satisfied in mild hypotheses, whereas local uniqueness and existence of geodesic coordinates must be required (the former is an open question even in manifolds with boundary, see [1]).0. Notation.1.By a manifold we shall always mean a smooth manifold; topologically it is a connected, paracompact space.For any