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Journal articles on the topic 'Compact smooth manifold'

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Published: 30 May 2022

Last updated: 31 May 2022

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1

CALCUT, J. S. "TORELLI ACTIONS AND SMOOTH STRUCTURES ON FOUR MANIFOLDS." Journal of Knot Theory and Its Ramifications 17, no. 02 (February 2008): 171–90. http://dx.doi.org/10.1142/s0218216508006075.

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Artin presentations are discrete equivalents of planar open book decompositions of closed, orientable three manifolds. Artin presentations characterize the fundamental groups of closed, orientable three manifolds. An Artin presentation also determines a smooth, compact, simply conected four manifold that bounds the three dimensional open book. In this way, the study of three and four manifolds may be approached purely group theoretically. In the theory of Artin presentations, elements of the Torelli subgroup act on the topology and smooth structures of the three and four manifolds. We show that the Torelli action can preserve the continuous topological type of a four manifold while changing its smooth structure. This is a new, group theoretic method of altering the smooth structure on a four manifold.

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2

Silver, Daniel S. "Examples of 3-knots with no minimal Seifert manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 3 (November 1991): 417–20. http://dx.doi.org/10.1017/s0305004100070481.

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We work throughout in the smooth category. Homomorphisms of fundamental and homology groups are induced by inclusion. Ann-knot, formn≥ 1, is an embeddedn-sphereK⊂Sn+2. ASeifert manifoldforKis a compact, connected, orientable (n+ 1)-manifoldV⊂Sn+2with boundary ∂V=K. By [9] Seifert manifolds always exist. As in [9] letYdenoteSn+2split alongV; Yis a compact manifold with ∂Y=V0∪V1, whereVt≈V. We say thatVis aminimal Seifert manifoldforKif π1Vt→ π1Yis a monomorphism fort= 0, 1. (Here and throughout basepoint considerations are suppressed.)

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3

JOST, JÜRGEN, and YI-HU YANG. "KÄHLER MANIFOLDS AND FUNDAMENTAL GROUPS OF NEGATIVELY δ-PINCHED MANIFOLDS." International Journal of Mathematics 15, no. 02 (March 2004): 151–67. http://dx.doi.org/10.1142/s0129167x04002247.

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In this note, we will show that the fundamental group of any negatively δ-pinched [Formula: see text] manifold cannot be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F4(-20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds [Formula: see text] of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for δ-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in SO(n,1)(n≥3) is the fundamental group of a quasi-compact Kähler manifold [21].

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4

Alpern, S., and V. Prasad. "End Behaviour and Ergodicity for Homeomorphisms of Manifolds with Finitely Many Ends." Canadian Journal of Mathematics 39, no. 2 (April 1, 1987): 473–91. http://dx.doi.org/10.4153/cjm-1987-020-5.

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The recent paper of Berlanga and Epstein [5] demonstrated the significant role played by the “ends” of a noncompact manifold M in answering questions relating homeomorphisms of M to measures on M. In this paper we show that an analysis of the end behaviour of measure preserving homeomorphisms of a manifold also leads to an understanding of some of their ergodic properties, and allows results previously obtained for compact manifolds to be extended (with qualifications) to the noncompact case. We will show that ergodicity is typical (dense Gδ) with respect to various compact-open topology closed subsets of the space consisting of all homeomorphisms of a manifold M which preserve a measure μ. It may be interesting for topologists to note that we prove when M is a σ-compact connected n-manifold, n≧ 2, then M is the countable union of an increasing family of compact connected manifolds. If M is a PL or smooth manifold, this is well known and easy. If M is just, however, a topological n-manifold then we apply the recent results [9] and [12] to prove the result. The Borel measure μ, is taken to be nonatomic, locally finite, positive on open sets, and zero for the manifold boundary of M.

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5

Katok, Anatole, and Keith Burns. "Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems." Ergodic Theory and Dynamical Systems 14, no. 4 (December 1994): 757–85. http://dx.doi.org/10.1017/s0143385700008142.

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AbstractWe establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

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6

Dancer, Andrew, and Andrew Swann. "The Geometry of Singular Quaternionic Kähler Quotients." International Journal of Mathematics 08, no. 05 (August 1997): 595–610. http://dx.doi.org/10.1142/s0129167x97000317.

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Two descriptions of quaternionic Kähler quotients by proper group actions are given: the first as a union of smooth manifolds, some of which come equipped with quaternionic Kähler or locally Kähler structures; the second as a union of quaternionic Kähler orbifolds. In particular the quotient always has an open set which is a smooth quaternionic Kähler manifold. When the original manifold and the group are compact, we describe a length space structure on the quotient. Similar descriptions of singular hyperKähler and 3-Sasakian quotients are given.

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7

Lee, Manseob. "Positively Continuum-Wise Expansiveness for C1 Differentiable Maps." Mathematics 7, no. 10 (October 16, 2019): 980. http://dx.doi.org/10.3390/math7100980.

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We show that if a differentiable map f of a compact smooth Riemannian manifold M is C 1 robustly positive continuum-wise expansive, then f is expanding. Moreover, C 1 -generically, if a differentiable map f of a compact smooth Riemannian manifold M is positively continuum-wise expansive, then f is expanding.

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8

Glöckner, Helge, and Alexander Schmeding. "Manifolds of mappings on Cartesian products." Annals of Global Analysis and Geometry 61, no. 2 (January 5, 2022): 359–98. http://dx.doi.org/10.1007/s10455-021-09816-y.

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AbstractGiven smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.

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9

Guan, Daniel. "Type II compact almost homogeneous manifolds of cohomogeneity one-II." International Journal of Mathematics 30, no. 13 (December 2019): 1940002. http://dx.doi.org/10.1142/s0129167x19400020.

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In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds [Formula: see text] obtained by blowing up the diagonal of the product of two copies of a [Formula: see text].

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10

Kołodziej, Sławomir, and Ngoc Cuong Nguyen. "Weak solutions of complex Hessian equations on compact Hermitian manifolds." Compositio Mathematica 152, no. 11 (September 9, 2016): 2221–48. http://dx.doi.org/10.1112/s0010437x16007417.

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We prove the existence of weak solutions of complex $m$-Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$, $p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

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11

Wang, Changyou. "Regularity and Blow-Up Analysis for J-Holomorphic Maps." Communications in Contemporary Mathematics 05, no. 04 (August 2003): 671–704. http://dx.doi.org/10.1142/s0219199703001063.

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If u∈H1(M,N) is a weakly J-holomorphic map from a compact without boundary almost hermitian manifold (M,j,g) into another compact without boundary almost hermitian manifold (N,J,h). Then it is smooth near any point x where Du has vanishing Morrey norm ℳ2,2m-2, with 2m= dim (M). Hence H2m-2measure of the singular set for a stationary J-holomorphic map is zero. Blow-up analysis and the energy quantization theorem are established for stationary J-holomorphic maps. Connections between stationary J-holomorphic maps and stationary harmonic maps are given for either almost Kähler manifolds M and N or symmetric ∇hJ.

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12

HEBEY, EMMANUEL, FRÉDÉRIC ROBERT, and YULIANG WEN. "COMPACTNESS AND GLOBAL ESTIMATES FOR A FOURTH ORDER EQUATION OF CRITICAL SOBOLEV GROWTH ARISING FROM CONFORMAL GEOMETRY." Communications in Contemporary Mathematics 08, no. 01 (February 2006): 9–65. http://dx.doi.org/10.1142/s0219199706002027.

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Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like Pgu = u2♯-1, where [Formula: see text] is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and [Formula: see text] is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

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13

FUCCI, GUGLIELMO, and KLAUS KIRSTEN. "THE CASIMIR EFFECT FOR GENERALIZED PISTON GEOMETRIES." International Journal of Modern Physics A 27, no. 15 (June 14, 2012): 1260008. http://dx.doi.org/10.1142/s0217751x12600081.

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In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type I ×f N where I = [a, b] is an interval of the real line and N is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at R ∈ (a, b). By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function f and base manifold N.

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14

FUCCI, GUGLIELMO, and KLAUS KIRSTEN. "THE CASIMIR EFFECT FOR GENERALIZED PISTON GEOMETRIES." International Journal of Modern Physics: Conference Series 14 (January 2012): 100–114. http://dx.doi.org/10.1142/s201019451200726x.

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In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type I ×f N where I = [a, b] is an interval of the real line and N is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at R ∈ (a, b). By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function f and base manifold N.

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15

SALUR, SEMA. "DEFORMATIONS OF SPECIAL LAGRANGIAN SUBMANIFOLDS." Communications in Contemporary Mathematics 02, no. 03 (August 2000): 365–72. http://dx.doi.org/10.1142/s0219199700000177.

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In [7], R. C. McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its dimension is equal to the dimension of ℋ1(L), the space of harmonic 1-forms on L. In this paper, we will show that the moduli space of all infinitesimal special Lagrangian deformations of L in a symplectic manifold with non-integrable almost complex structure is also a smooth manifold of dimension b1(L), the first Betti number of L.

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16

Hasui, Sho, Hideya Kuwata, Mikiya Masuda, and Seonjeong Park. "Classification of Toric Manifolds over an n-Cube with One Vertex Cut." International Mathematics Research Notices 2020, no. 16 (July 5, 2018): 4890–941. http://dx.doi.org/10.1093/imrn/rny161.

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Abstract A complete nonsingular toric variety (called a toric manifold) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\operatorname{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda’s three-fold, the simplest non-projective toric manifold, is over $\operatorname{vc}(I^3)$. In this paper, we classify toric manifolds over $\operatorname{vc}(I^n)$$(n\ge 3)$ as varieties and as smooth manifolds. It consequently turns out that there are many non-projective toric manifolds over $\operatorname{vc}(I^n)$ but they are all diffeomorphic, and toric manifolds over $\operatorname{vc}(I^n)$ in some class are determined by their cohomology rings as varieties.

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17

Walczak, Paweł G. "Dynamics of the geodesic flow of a foliation." Ergodic Theory and Dynamical Systems 8, no. 4 (December 1988): 637–50. http://dx.doi.org/10.1017/s0143385700004740.

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AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

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18

Boyer, Charles P., and Krzysztof Galicki. "The Twistor Space of a 3-Sasakian Manifold." International Journal of Mathematics 08, no. 01 (February 1997): 31–60. http://dx.doi.org/10.1142/s0129167x97000032.

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Any compact 3-Sasakian manifold [Formula: see text] is a principal circle V-bundle over a compact complex orbifold [Formula: see text]. This orbifold has a contact Fano structure with a Kähler–Einstein metric of positive scalar curvature and it is the twistor space of a positive compact quaternionic Kähler orbifold [Formula: see text]. We show that many results known to hold when [Formula: see text] is a smooth manifold extend to this more general singular case. However, we construct infinite families of examples with [Formula: see text] which sharply differs from the smooth case, where there is only one such [Formula: see text].

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19

Cherepanov, V. V. "Orbit spaces for torus actions on Hessenberg varieties." Sbornik: Mathematics 212, no. 12 (December 1, 2021): 1765–84. http://dx.doi.org/10.1070/sm9278.

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Abstract In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.

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20

Cherepanov, V. V. "Orbit spaces for torus actions on Hessenberg varieties." Sbornik: Mathematics 212, no. 12 (December 1, 2021): 1765–84. http://dx.doi.org/10.1070/sm9278.

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Abstract In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.

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21

TILLMANN, ULRIKE. "Homology stability for symmetric diffeomorphism and mapping class groups." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (December 2, 2015): 121–39. http://dx.doi.org/10.1017/s0305004115000638.

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AbstractFor any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

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22

Deshmukh, Sharief, and Olga Belova. "On Killing Vector Fields on Riemannian Manifolds." Mathematics 9, no. 3 (January 28, 2021): 259. http://dx.doi.org/10.3390/math9030259.

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We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

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23

ATIYAH, MICHAEL, and CLAUDE LEBRUN. "Curvature, cones and characteristic numbers." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (April 25, 2013): 13–37. http://dx.doi.org/10.1017/s0305004113000169.

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AbstractWe study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulæ to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.

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24

LEE, KYUNG BAI, and FRANK RAYMOND. "MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES." International Journal of Mathematics 02, no. 01 (February 1991): 67–76. http://dx.doi.org/10.1142/s0129167x91000065.

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Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

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25

Poddar, Mainak, and Ajay Singh Thakur. "Group actions, non-Kähler complex manifolds and SKT structures." Complex Manifolds 5, no. 1 (February 2, 2018): 9–25. http://dx.doi.org/10.1515/coma-2018-0002.

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AbstractWe give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.

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Afanas'eva, Olena, and Viktoriia Bilet. "On a connection between some classes of mapping on Riemannian manifolds." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 34 (April 24, 2021): 3–10. http://dx.doi.org/10.37069/1683-4720-2020-34-1.

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In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.

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27

LEE, JUNG CHAN, JEONG HYEONG PARK, and KOUEI SEKIGAWA. "CRITICAL HERMITIAN STRUCTURES ON THE PRODUCT OF SASAKIAN MANIFOLDS." International Journal of Geometric Methods in Modern Physics 09, no. 07 (September 7, 2012): 1250055. http://dx.doi.org/10.1142/s0219887812500557.

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Let [Formula: see text] be a compact orientable smooth manifold admitting an almost complex structure and [Formula: see text] for (λ, μ) ∈ ℝ2 - (0, 0) be the functional defined on the space of the almost Hermitian structure [Formula: see text]. We discuss the first variational problem of the functional [Formula: see text] on the space [Formula: see text] and its subspace [Formula: see text] in the case where [Formula: see text] is a product manifold of Sasakian manifolds. Further this paper provides examples of critical Hermitian structures of the functional [Formula: see text] for various (λ, μ).

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LÉANDRE, RÉMI. "STOCHASTIC ADAMS THEOREM FOR A GENERAL COMPACT MANIFOLD." Reviews in Mathematical Physics 13, no. 09 (September 2001): 1095–133. http://dx.doi.org/10.1142/s0129055x01000909.

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We give a stochastic analoguous of the theorem of Adams, which says that the Hochschild cohomology is equal to the cohomology of the based smooth loop space. The key tools are the stochastic Chen iterated integrals as well as Driver's flow.

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29

BAKLOUTI, ALI, SAMI DHIEB, and KHALED TOUNSI. "WHEN IS THE DEFORMATION SPACE $\mathscr{T}(\Gamma, H_{2n+1}, H)$ A SMOOTH MANIFOLD?" International Journal of Mathematics 22, no. 11 (November 2011): 1661–81. http://dx.doi.org/10.1142/s0129167x11007331.

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Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

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30

Kim, Yoosik, Jaeho Lee, and Fumihiko Sanda. "Detecting non-displaceable toric fibers on compact toric manifolds via tropicalizations." International Journal of Mathematics 30, no. 01 (January 2019): 1950003. http://dx.doi.org/10.1142/s0129167x19500034.

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We provide a combinatorial way to locate non-displaceable Lagrangian toric fibers on any compact toric manifold. By taking the intersection of certain tropicalizations coming from its moment polytope, one can detect all Lagrangian toric fibers having non-vanishing Floer cohomology ([K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151(1) (2010) 23–174; K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17(3) (2011) 609–711.]). The intersection completely characterizes all non-displaceable toric fibers, in some cases including pseudo symmetric smooth Fano varieties ([G. Ewald, On the classification of toric Fano varieties, Discrete Comput. Geom. 3(1) (1988) 49–54.]).

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31

BEN HALIMA, MAJDI. "CONSTRUCTION OF CERTAIN FUZZY FLAG MANIFOLDS." Reviews in Mathematical Physics 22, no. 05 (June 2010): 533–48. http://dx.doi.org/10.1142/s0129055x10004016.

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Approximating the algebra of complex-valued smooth functions on a space-time manifold by a sequence of matrix algebras [Formula: see text], with dN ↗ ∞, is the basic idea of fuzzy manifolds. In this paper, we explicitly construct fuzzy versions of the homogeneous spaces SO(2n + 1)/U(n) and Sp (n)/U(1) × Sp (n - 1) for n ≥ 2. This allows us to extend a result of Zhang giving a construction of fuzzy irreducible compact Hermitian symmetric spaces to a class of flag manifolds.

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32

Papadima, Stefan, and Alexander I. Suciu. "The Topology of Compact Lie Group Actions Through the Lens of Finite Models." International Mathematics Research Notices 2019, no. 20 (January 29, 2018): 6390–436. http://dx.doi.org/10.1093/imrn/rnx294.

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AbstractGiven a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold, which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the representation varieties of π1(M). As an application, we show that compact Sasakian manifolds of dimension 2n + 1 are (n − 1)-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.

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33

Liu, Pei-Dong. "Pesin’s entropy formula for endomorphisms." Nagoya Mathematical Journal 150 (June 1998): 197–209. http://dx.doi.org/10.1017/s0027763000025113.

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DEMYANOVICH, YU K. "SPLINE APPROXIMATIONS ON MANIFOLDS." International Journal of Wavelets, Multiresolution and Information Processing 04, no. 03 (September 2006): 383–403. http://dx.doi.org/10.1142/s0219691306001324.

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A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).

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35

EGASHIRA, SHINJI. "Qualitative theory and expansion growth of transversely piecewise-smooth foliated ${S}^{1}$-bundles." Ergodic Theory and Dynamical Systems 17, no. 2 (April 1997): 331–47. http://dx.doi.org/10.1017/s0143385797079157.

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We study a qualitative theory of compact, transversely piecewise-smooth foliated $S^1$-bundles. We show that it has the same qualitative properties as that of smooth codimension-one foliations on a compact manifold. By the obtained qualitative properties, we can deduce that the expansion growth of this foliation is classified and that the entropy of this foliation is positive if and only if there exists a resilient leaf.

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36

Humbert, Emmanuel. "Best constants in the L2-Nash inequality." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 3 (June 2001): 621–46. http://dx.doi.org/10.1017/s0308210500001025.

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37

Micheletti, Anna Maria, and Angela Pistoia. "On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds." International Journal of Differential Equations 2010 (2010): 1–11. http://dx.doi.org/10.1155/2010/432759.

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38

Matsuoka, Takashi. "The number of periodic points of smooth maps." Ergodic Theory and Dynamical Systems 9, no. 1 (March 1989): 153–63. http://dx.doi.org/10.1017/s0143385700004879.

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AbstractLetf: M → M be a C1 map on a compact manifold. We give a topological condition under which f has an even number of periodic points with a given period. We also obtain a sufficient condition, in terms of homology, for ƒ to have infinitely many periodic points.

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39

Dumas, H. S., F. Golse, and P. Lochak. "Multiphase averaging for generalized flows on manifolds." Ergodic Theory and Dynamical Systems 14, no. 1 (March 1994): 53–67. http://dx.doi.org/10.1017/s0143385700007720.

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AbstractWe present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in Rm and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.

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van den Berg, M., and P. Gilkey. "The heat content asymptotics of a time-dependent process." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 2 (April 2000): 307–12. http://dx.doi.org/10.1017/s0308210500000172.

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XIA, QIAOLING. "ON THE FIRST EIGENCONE FOR THE FINSLER LAPLACIAN." Bulletin of the Australian Mathematical Society 94, no. 2 (March 16, 2016): 316–25. http://dx.doi.org/10.1017/s0004972716000034.

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Beals, Richard, and Nancy K. Stanton. "The Heat Equation for the -Neumann Problem, II." Canadian Journal of Mathematics 40, no. 2 (April 1, 1988): 502–12. http://dx.doi.org/10.4153/cjm-1988-021-8.

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Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

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Jbilou, Asma. "Complex Hessian Equations on Some Compact Kähler Manifolds." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–48. http://dx.doi.org/10.1155/2012/350183.

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On a compact connected2m-dimensional Kähler manifold with Kähler formω, given a smooth functionf:M→ℝand an integer1<k<m, we want to solve uniquely in[ω]the equationω̃k∧ωm-k=efωm, relying on the notion ofk-positivity forω̃∈[ω](the extreme cases are solved:k=mby (Yau in 1978), andk=1trivially). We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

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CAO, JIANGUO, and MEI-CHI SHAW. "THE SMOOTHNESS OF RIEMANNIAN SUBMERSIONS WITH NON-NEGATIVE SECTIONAL CURVATURE." Communications in Contemporary Mathematics 07, no. 01 (February 2005): 137–44. http://dx.doi.org/10.1142/s0219199705001672.

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Let Mn be a complete, non-compact and C∞-smooth Riemannian manifold with non-negative sectional curvature. Suppose that [Formula: see text] is a soul of Mn given by the fundamental theory of Cheeger and Gromoll, and suppose that [Formula: see text] is a distance non-increasing retraction from the whole manifold to the soul (e.g. the retraction given by Sharafutdinov). Then we show that the retraction Ψ above must give rise to a C∞-smooth Riemannian submersion from Mn to the soul [Formula: see text]. Moreover, we derive a new flat strip theorem associated with the Cheeger–Gromoll convex exhaustion for the manifold above.

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Gu, Shijie, and Craig R. Guilbault. "Compactifications of manifolds with boundary." Journal of Topology and Analysis 12, no. 04 (December 31, 2018): 1073–101. http://dx.doi.org/10.1142/s1793525319500754.

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This paper is concerned with compactifications of high-dimensional manifolds. Siebenmann’s iconic 1965 dissertation [L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton Univ. (1965), MR 2615648] provided necessary and sufficient conditions for an open manifold [Formula: see text] ([Formula: see text]) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where [Formula: see text] is noncompact with compact boundary; however, when [Formula: see text] is noncompact, the situation is more complicated. The goal becomes a “completion” of [Formula: see text], i.e. a compact manifold [Formula: see text] containing a compactum [Formula: see text] such that [Formula: see text]. Siebenmann did some initial work on this topic, and O’Brien [G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl. 16 (1983) 303–324, MR 722123] extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here, we provide such a characterization. Our second main theorem involves [Formula: see text]-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann [T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171–208, MR 0425973] guarantee [Formula: see text]-compactifiability for a manifold [Formula: see text]. We cannot answer that question, but we do show that those conditions are satisfied if and only if [Formula: see text] is [Formula: see text]-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem — an application that partly explains our current interest in that topic, and also illustrates the utility of the [Formula: see text]-condition found in that theorem.

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LI, YANYAN, and MEIJUN ZHU. "YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS." Communications in Contemporary Mathematics 01, no. 01 (February 1999): 1–50. http://dx.doi.org/10.1142/s021919979900002x.

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A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

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47

Gauld, David. "Manifold Neighbourhoods and a Conjecture of Adjamagbo." New Zealand Journal of Mathematics 52 (September 19, 2021): 167–74. http://dx.doi.org/10.53733/131.

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We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.

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48

Burns, Keith, and Marlies Gerber. "Continuous invariant cone families and ergodicity of flows in dimension three." Ergodic Theory and Dynamical Systems 9, no. 1 (March 1989): 19–25. http://dx.doi.org/10.1017/s014338570000479x.

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AbstractIt is shown that a C2 flow on a compact three-dimensional manifold that preserves a smooth measure and has a continuous family of cones satisfying a certain invariance property must be ergodic.

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49

Хохлюк, Олександра Олександрівна, and Sergiy Ivanovych Maksymenko. "Smooth approximations and their applications to homotopy types." Proceedings of the International Geometry Center 13, no. 2 (September 13, 2020): 68–108. http://dx.doi.org/10.15673/tmgc.v13i2.1781.

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Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It is also established a parametrized variant of such a result.In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

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Lee, Keon-Hee, and Jong-Myung Kim. "Hyperbolic manifolds with the strongly shadowing property." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 37–43. http://dx.doi.org/10.1017/s000497270003330x.

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Let f be a C1 diffeomorphism of a compact smooth manifold M and λ ⊂ M a C1 compact invariant submanifold with a hyperbolic structure as a subset of M. We show that the diffeomorphism | λ is Anosov if and only if λ has the strongly shadowing property, and find hyperbolic sets which have the strongly shadowing property.

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