N-dimensional Manifold Research Articles

Stability estimates for an inverse Steklov problem in a class of hollow spheres

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem.

Compactness for conformal scalar-flat metrics on umbilic boundary manifolds

Let (M,g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well known that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provided n=8 and the Weyl tensor of the boundary is always different from zero, or if n>8 and the Weyl tensor of M is always different from zero on the boundary.

Ledoux-type rigidity results on manifolds with boundary in the presence of symmetries

We consider a 3-dimensional complete Riemannian manifold (M,g) with boundary and assuming that an optimal Sobolev or Hardy–Sobolev inequality holds with the same best constant as in the case of the solid torus T we investigate the relationship between the geometry of such a manifold and that of the solid torus and we prove Ledoux-type rigidity results (see in Ledoux (1999)) giving a positive answer to a question posed to us by the referee of our recently published paper Cotsiolis and Labropoulos (2018). Some generalization results, for n-dimensional Riemannian manifolds (M,g) with n>3, are also presented.

D-homothetic deformation of an h-Einstein Para-Sasakian manifold

Objectives: The main purpose of this paper is to study the theory of Dhomothetic deformation of an h-Einstein Para-Sasakian manifold. Methods: A deformation technique is employed to solve the resulting problem. We provide its application in the general theory of relativity. Findings: Section 2 deals with recurrent and symmetric Para-Sasakian Manifolds. In section 3 we have defined and studied projectively symmetric Para-Sasakian manifold. The notion of ϕ -holomorphic sectional curvature in an h-Einstein Para-Sasakian manifold has been delineated in the section 4. In this very section, we shall study the relevant commutation formulae which give rise to the required curvature tensor. The obtained results were compared with the previous works(1--3) in this field and were found to be in good agreement. Novelty/Conclusion: The study concludes the following results: 1. If n-dimensional Para-Sasakian manifold is to be recurrent projective symmetric then the recurrent projective curvature tensor vanishes identically.2. If n-dimensional Para-Sasakian manifold is to be projective semi-symmetric then the semi-recurrent projective curvature tensor vanishes identically.3. If n-dimensional Para-Sasakian manifold is to be projective scalar-symmetric then the scalar projective curvature vanishes identically.4. If D-homothetic deformation of h-Einstein Para-Sasakian manifold is constant ϕ -holomorphic sectional curvature then we shall obtain the value of scalar fields. Keywords: ϕ -holomorphic sectional curvature; constant curvature; Para-Sasakian manifold; recurrent projective symmetric; D-homothetic deformation; h-Einstein manifold

Besse conjecture for compact manifolds with pinched curvature

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

Fundamental tones of clamped plates in nonpositively curved spaces

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M,g) is universally bounded from below by (n−1)416κ4 whenever the κ-Cartan-Hadamard conjecture holds on (M,g), e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in (M,g) of volume v>0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature −κ2 provided that v≤cn/κn with c2≈21.031 and c3≈1.721, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. K≡κ=0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low-dimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on (M,g)) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

Extrinsic upper bound of the eigenvalue for [formula omitted]-Laplacian

Let M be an m-dimensional closed orientable submanifold in an n-dimensional Riemannian manifold N. When the sectional curvature of N is bounded above by δ, we obtain an upper bound for the first nonzero eigenvalue of the p-Laplacian in terms of the second fundamental form of M and δ. This generalizes the Reilly-type inequality for the Laplacian (Heintze, 1988; Reilly, 1977) to the p-Laplacian and extends the work of Chen and Wei (2019) and Du and Mao (2015) for the p-Laplacian.

Doubling chains on complements of algebraic hypersurfaces

A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ: B1 → Y of the unit ball in ℂn, which is extendable to the (say) four times larger concentric ball of B1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney’s ball coverings of a domain W ⊂ ℝn, introduced in [19] (compare [11]). We study doubling chains in the complement Y = ℂnH of a complex algebraic hypersurface H of degree d in ℂn, and provide information on their length and other properties. Our main result is that any two points v1, v2 in a distance δ from H can be joined via a doubling chain in the complement Y = ℂnH of length at most c1$$\log \left( {\frac{{{c_2}}}{\delta }} \right)$$ with explicit constants c1, c2 depending only on n and d. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.

Some Examples of Projective and c-projective Compactifications of Einstein Metrics

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi–Hanson instanton admits a projective compactification which is non-metric, and that a metric cone over any (pseudo)-Riemannian manifolds admits a metric projective compactification. We construct a para-c-projective compactification of neutral signature Einstein metrics canonically defined on certain rank-n affine bundles M over n-dimensional manifolds endowed with projective structures.

Global existence of smooth solutions for wave maps in de Sitter spacetime

In this paper, we study the problem of a harmonic map from de Sitter spacetime into a geodesic on an n-dimensional complete Riemannian manifold. We show that the global smooth solution exists for such wave map when the solution is a small perturbation along the geodesic solution by the method of weighted energy estimates.

Obstruction Class for the Existence of a Conformal Spin Structure in a Strict Sense

Let V be a pseudo-Riemannian n-dimensional manifold or, more generally, let $$(\xi ,Q)$$ be a real fibre bundle whose base space is a paracompact space endowed with a non-degenerate quadratic form Q, (that is, with a structure group $${\mathrm {O}}(p,q),$$ $$n=p+q$$.) Let $$K_{p,q}$$ denote the obstruction class for the existence of a $$\mathrm {Pin}(p,q)$$-spin structure on V or over $$\xi .$$ Let $$K_{\mathrm {Conf}}(p,q)$$ denote the obstruction class for the existence of a conformal spin structure in a strict sense on V or over $$\xi ,$$ (simply: a $$C_{n}^{s}(p,q)$$-spin structure), if $$n=2r,$$ or of a conformal special spin structure, if $$n=2r+1.$$ This short self-contained paper will recall the determination of the obstruction class $$K_{p,q}$$ on V, or over $$\xi ,$$ for n even or odd. Then, the obstruction class $$K_{p+1,q+1}$$ for the existence of a $$\mathrm {Pin}(p+1,q+1)$$-spin structure over $$\xi _{j}$$, (Greub’s j-extension of $$\xi ,$$ where j denotes the identity mapping from $${\mathrm {O}}(p,q)$$ into $${\mathrm {O}}(p+1,q+1)$$), will be determined in order to express $$K_{\mathrm {Conf}}(p,q),$$ for $$n=2r$$ or $$n=2r+1,$$ in terms of the Stiefel–Whitney classes $$w_{i}(p,q),$$ $$i=1,2,$$ of $$\xi $$, decomposed as the Whitney sum $$\xi =\xi ^{+}\oplus \xi ^{-},$$ where the restriction of Q to $$\xi ^{+}$$ is positive definite and the restriction of Q to $$\xi ^{-}$$ is negative. If $$n=2r,$$ we find again results obtained in previous publications [4, 5, 7], by different methods.

Some rigidity characterizations on critical metrics for quadratic curvature functionals

We study closed n-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifolds. Moreover, under some curvature conditions, the result that a critical metric must be Einstein is proved.

An elliptic equation on n-dimensional manifolds

We consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry of the manifold. We produce a nontrivial solution by using a Palais–Smale compactness condition and a mountain pass geometry.

On the geometry of the tangent bundle with vertical rescaled generalized Cheeger- Gromoll metric

Let (M, g) be an n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G f and called the vertical rescaled generalized Cheeger-Gromoll metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (TM, G f ).

*-Critical point equation on N(k)-contact manifolds

The object of the present paper is to characterize N(k)-contact metric manifolds satisfying the *-critical point equation. It is proved that, if (g, λ) is a non-constant solution of the *-critical point equation of a non-compact N(k)-contact metric manifold, then (1) the manifold M is locally isometric to the Riemannian product of a at (n + 1)-dimensional manifold and an n-dimensional manifold of positive curvature 4 for n > 1 and at for n = 1, (2) the manifold is *-Ricci at and (3) the function λ is harmonic. The result is also verified by an example.

Gromov-hyperbolicity and transitivity of geodesic flows in n-dimensional Finsler manifolds

We show that the geodesic flow of a compact Finsler manifold without conjugate points is transitive provided that the universal covering satisfies the uniform Finsler visibility condition. This result is a nontrivial extension of a well known theorem due to Eberlein for Riemannian manifolds. For doing so, we introduce suitable Finsler versions of the concepts of Gromov's δ-hyperbolicity and Eberlein's visibility, and study their consequences.

Центрированные плоскости в пространстве про­ективной связности

The space of centered planes is considered in the Cartan projec­ti­ve connection space . The space is important because it has con­nec­tion with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle. The space is an n-dimensional differentiable manifold with each point of which an n-dimensional projective space containing this point is associated. Thus, the manifold is the base, and the space is the n-dimensional fiber “glued” to the points of the base. A projective space is a quotient space of a linear space with respect to the equivalence (collinearity) of non-zero vectors, that is . The projective space is a manifold of di­men­sion n. In this paper we use the Laptev — Lumiste invariant analytical meth­od of differential geometric studies of the space of centered planes and introduce a fundamental-group connection in the associated bundle . The bundle contains four quotient bundles. It is show that the connection object is a quasi-tensor containing four subquasi-tensors that define connections in the corresponding quotient bundles.

Symplectic and Contact Geometry with Complex Manifolds

In this paper, we discuss about almost complex structures and complex structures on Riemannian manifolds, symplectic manifolds and contact manifolds. We have also shown a special comparison between complex symplectic geometry and complex contact geometry. Also, the existence of a complex submanifold of n-dimensional complex manifold which intersects a real submanifold
 GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 119-126

The Hausdorff Measure on n-Dimensional Manifolds in ℝm and n-Dimensional Variations

We extend the notion of the variation Vf([a; b]) of a function f : [a; b] → ℝ to the variation Vf(A) of a continuous map f : G → ℝn, where G is an open subset of ℝn, over a set A ⊂ G of the form A = ∪i ∈ IKi where I is countable and all Ki are compact. Let f : G → ℝm where G ⊂ ℝn with n ≤ m, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < in ≤ m, let fα be the map with coordinate functions $$ {f}_{i_1},\dots, {f}_{i_n} $$. The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝn, and a subset A ⊂ G has the form A = ∪i ∈ IKi where I is countable and all Ki are compact, then $$ {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) $$ where $$ {V}_{f_{\alpha }}(A) $$ is the variation of fα over A and Hn is the n-dimensional Hausdorff measure in ℝm.

Maclaurin Heat Coefficients and Associated Zeta Functions on Quaternionic Projective Spaces Pn(H) (n ≥ 1)

The heat invariants or the Minakshisundaram-Pleijel heat coefficients (M) (k ≥ 0) describe the asymptotic expansion of the heat kernel HM on any N = 4 n-dimensional (n ≥ 1) compact Riemannian manifold M; associated with the coefficients is the Minakshisundaram-Pleijel zeta function ζM = ζM(s) (s ∈ C). In this paper, we introduce and study a new class of heat coefficients, namely, the Maclaurin heat coefficients (t > 0 , m 0) (i.e., the coefficients appearing in the Maclaurin expansion of the heat kernel HM(t, θ)) in terms of the classical and generalised Minakshisundaram-Pleijel coefficients and (M) (0 ⩽ j ⩽ m) respectively, when M = P n(H) (n ≥ 1), a quaternionic projective space. Remarkable asymptotic expansions for the Maclaurin spectral functions are established. We also introduce and construct new zeta functions (m ≥ 0) associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions can be explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.