Some Characterizations of Generalized Manifolds With Boundaries

1952 ◽  
Vol 4 ◽  
pp. 329-342 ◽  
Author(s):  
Paul A. White
Keyword(s):  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Generalized Manifolds ◽  
Definition Of

In R. L. Wilder's book [2] the open and closed generalized manifolds are extensively studied. However, no study is made of the generalized manifold with boundary nor is a definition of such a space given except in the case of the generalized closed n-cell. A definition of a generalized manifold with boundary was given by the author in his paper [1]. Before undertaking the study of further properties of these manifolds it seems appropriate to characterize the manifolds with boundary in terms of the open and closed manifolds of Wilder. It is to that purpose that this paper is directed and in particular the generalized closed n-cell of Wilder is characterized as a special manifold with boundary.

scholarly journals Embedding Topological $n$-Manifolds into Compact Nonstandard Topological $n$-Manifolds with Boundary

2021 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Topological Manifold ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Topological Manifolds ◽  
One Point Compactification ◽  
Main Message

We show as a main message that there is a simple dimension-preserving way to openly and densely embed every topological manifold into a compact ``nonstandard'' topological manifold with boundary.This class of ``nonstandard'' topological manifolds with boundary contains the usual topological manifolds with boundary.In particular,the Alexandroff one-point compactification of every given topological $n$-manifold is a ``nonstandard'' topological $n$-manifold with boundary.

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary

2010 ◽  
Vol 53 (4) ◽  
pp. 674-683 ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Riemannian Manifolds ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Manifolds With Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Manifold With Boundary

AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

scholarly journals Unbounded Fredholm Operators and Spectral Flow

2005 ◽  
Vol 57 (2) ◽  
pp. 225-250 ◽  
Author(s):  
Bernhelm Booss-Bavnbek ◽  
Matthias Lesch ◽  
John Phillips
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Direct Methods ◽  
Spectral Flow ◽  
Surprising Result ◽  
Graph Distance ◽  
Manifolds With Boundary ◽  
Fredholm Operators ◽  
Homotopy Invariance ◽  
Definition Of

AbstractWe study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transformand direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

scholarly journals A long neck principle for Riemannian spin manifolds with positive scalar curvature

2020 ◽  
Vol 30 (5) ◽  
pp. 1183-1223
Author(s):  
Simone Cecchini
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Pairwise Disjoint ◽  
Index Theory ◽  
Spin Manifold ◽  
Manifolds With Boundary ◽  
Topological Information ◽  
Manifold With Boundary ◽  
Spin Manifolds ◽  
Nonzero Degree

AbstractWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$ scal ( X ) ≥ n ( n - 1 ) and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$ f : X → S n which is strictly area decreasing, then the distance between the support of $$\text {d}f$$ d f and the boundary of X is at most $$\pi /n$$ π / n . This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$ scal ( X ) > σ > 0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $$\partial X$$ ∂ X is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$ π ( n - 1 ) / ( n σ ) . Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$ N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$ scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$ 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.

scholarly journals Nilpotent quotients of fundamental groups of special 3-manifolds with boundary

1998 ◽  
Vol 58 (2) ◽  
pp. 233-237
Author(s):  
Gabriela Putinar
Keyword(s):  
Fundamental Group ◽  
Betti Number ◽  
Fundamental Groups ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Maximal Torsion ◽  
Torsion Free

We use a Betti number estimate of Freedman-Hain-Teichner to show that the maximal torsion-free nilpotent quotient of the fundamental group of a 3-manifold with boundary is either Z or Z ⊕ Z. In particular we reobtain the Evans-Moser classification of 3-manifolds with boundary which have nilpotent fundamental groups.

Integration of Equivariant Forms

2020 ◽  
pp. 228-232
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Differential Form ◽  
Circle Action ◽  
Manifolds With Boundary ◽  
Boundary Points ◽  
Manifold With Boundary ◽  
Lie Group Action ◽  
Cartan Model ◽  
Interior Points

This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with coefficients that are invariant forms on M. Such a form can be integrated by integrating the coefficients. This can be called equivariant integration. The chapter shows that under equivariant integration, Stokes's theorem still holds. Everything done so far in this book concerning a Lie group action on a manifold can be generalized to a manifold with boundary. An important fact concerning manifolds with boundary is that a diffeomorphism of a manifold with boundary takes interior points to interior points and boundary points to boundary points.

scholarly journals Trisections of 4-manifolds with boundary

2018 ◽  
Vol 115 (43) ◽  
pp. 10861-10868 ◽  
Author(s):  
Nickolas A. Castro ◽  
David T. Gay ◽  
Juanita Pinzón-Caicedo
Keyword(s):  
Existence Result ◽  
Manifolds With Boundary ◽  
Open Book ◽  
Original Proof ◽  
Open Book Decomposition ◽  
Manifold With Boundary ◽  
The Given

Given a handle decomposition of a 4-manifold with boundary and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components and show how trisected 4-manifolds with multiple boundary components glue together.

Existence of a Closed Geodesic on Non-compact Riemannian Manifolds with Boundary

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Germinario ◽  
Miguel Sánchez
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesic ◽  
Detailed Comparison ◽  
Manifolds With Boundary ◽  
Manifold With Boundary

AbstractA new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

scholarly journals About the Uniqueness of Conformal Metrics with Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundary

2011 ◽  
Vol 13 ◽  
pp. 71-79
Author(s):  
Gonzalo García ◽  
Jhovanny Muñoz
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Manifolds With Boundary ◽  
Conformal Metrics ◽  
Dirichlet Eigenvalue ◽  
Uniqueness Results ◽  
Manifold With Boundary ◽  
First Dirichlet Eigenvalue ◽  
Neumann Eigenvalue ◽  
Linear Elliptic Problem

Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

scholarly journals Logarithms and sectorial projections for elliptic boundary problems

2008 ◽  
Vol 103 (2) ◽  
pp. 243 ◽  
Author(s):  
Anders Gaarde ◽  
Gerd Grubb
Keyword(s):  
Compact Manifold ◽  
Elliptic Boundary ◽  
Zeta Functions ◽  
Boundary Problems ◽  
Wodzicki Residue ◽  
Manifold With Boundary ◽  
Special Cases ◽  
The Difference ◽  
Definition Of ◽  
Spectral Cut

On a compact manifold with boundary, consider the realization $B$ of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say $\mathsf{R}_{-}$. In the first part of the paper we define and discuss in detail the operator $\log B$; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, $\zeta (B,0)$, and it enters in an important way in studies of composed zeta functions $\zeta (A,B,s)= {\operatorname {Tr}}(AB^{-s})$ (pursued elsewhere). There is a similar definition of the operator $\log_{\theta}B$, when the spectral cut is at a general angle $\theta$. When $B$ has spectral cuts at two angles $\theta <\varphi$, one can define the sectorial projection $\Pi_{\theta,\varphi} (B)$ whose range contains the generalized eigenspaces for eigenvalues with argument in $\left]\theta,\varphi \right[$; this is studied in the last part of the paper. The operator $\Pi_{\theta ,\varphi}(B)$ is shown to be proportional to the difference between $\log_{\theta}B$ and $\log_{\varphi} B$, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.

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