Geometric Properties Of Manifolds Research Articles

Jordan Groups and Geometric Properties of Manifolds

Jordan Groups and Geometric Properties of Manifolds

Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in manifold learning. In this paper, we provide a positive answer to the problem. Specifically, we use eigenvector fields to construct local coordinate charts with low distortion, and show that the distortion constants depend only on geometric properties of manifolds with metrics in the little Holder space $$c^{2,\alpha }$$ . Next, we use the coordinate charts to embed the entire manifold into a finite dimensional Euclidean space. The proof of the results relies on solving the elliptic system and providing estimates for eigenvector fields and the heat kernel and their gradients. We also provide approximation results for eigenvector field under the $$c^{2,\alpha }$$ perturbation.

Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form

We study local geometric properties of manifolds equipped with a closed 2-form nondegenerate at all points of a dense proper subset. We introduce the natural notion of tame singular point, at which the matrix of the 2-form degenerates in a regular way. We find a condition for Hamiltonian dynamical systems to be extended smoothly to tame singular points, generalize the Darboux theorem about the local reduction of the matrix of the 2-form to canonical form, and study the singular behavior of directional gradients.

Classical Differential Equations on Manifolds

1. Types of differential objects on manifolds. Let D be a differential object, say, the Laplace potential partial differential equation, the wave equation, the diffusion equation, or one of the corresponding differential operators defined in the Cartesian coordinates of Euclidean n-space Rn. We shall say that D is well-defined on a differentiable manifold Mn (connected, separable metric space with C* local coordinates) in case there is a differential equation or a differential operator defined on Mn, which in a certain atlas (covering of Mn by a subcollection of the differentiable coordinate systems) is expressed by D. The collection of all differentiable homeomorphisms of open sets of Rn into R , which preserve D, is the pseudogroup of D, cf. [II]. Then we can state that Mn admits the type of differential object D if and only if Mn has an atlas whose coordinate transition maps belong to the pseudogroup of D. In this paper we shall find topological and geometrical properties of manifolds which admit the classical differential objects listed above. In the next section we relate this problem to the more standard one of reducing the structure group of the principal bundle of bases B(Mn). In section three we study applications of the general theory to the Laplace and wave partial differential equations and also to linear ordinary differential equations with constant coefficients. Finally, in the appendix we state properties of bundle reductions to totally disconnected subgroups; these results being immediate generalizations of an earlier work of the authors [5] and of relevance for the present study. 2. Bundle reduction for the potential, wave, diffusion equations and operators. We note that the pseudogroup for the potential operator

Classical differential equations on manifolds

1. Types of differential objects on manifolds. Let D be a differential object, say, the Laplace potential partial differential equation, the wave equation, the diffusion equation, or one of the corresponding differential operators defined in the Cartesian coordinates of Euclidean n-space Rn. We shall say that D is well-defined on a differentiable manifold Mn (connected, separable metric space with C* local coordinates) in case there is a differential equation or a differential operator defined on Mn, which in a certain atlas (covering of Mn by a subcollection of the differentiable coordinate systems) is expressed by D. The collection of all differentiable homeomorphisms of open sets of Rn into R , which preserve D, is the pseudogroup of D, cf. [II]. Then we can state that Mn admits the type of differential object D if and only if Mn has an atlas whose coordinate transition maps belong to the pseudogroup of D. In this paper we shall find topological and geometrical properties of manifolds which admit the classical differential objects listed above. In the next section we relate this problem to the more standard one of reducing the structure group of the principal bundle of bases B(Mn). In section three we study applications of the general theory to the Laplace and wave partial differential equations and also to linear ordinary differential equations with constant coefficients. Finally, in the appendix we state properties of bundle reductions to totally disconnected subgroups; these results being immediate generalizations of an earlier work of the authors [5] and of relevance for the present study. 2. Bundle reduction for the potential, wave, diffusion equations and operators. We note that the pseudogroup for the potential operator