On $G$-sets and isospectrality

Ori Parzanchevski

Displaying similar documents to “On $G$ -sets and isospectrality”

Collapse of warped submersions

Szymon M. Walczak (2006)

Annales Polonici Mathematici

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We generalize the concept of warped manifold to Riemannian submersions π: M → B between two compact Riemannian manifolds $(M, g_{M})$ and $(B, g_{B})$ in the following way. If f: B → (0,∞) is a smooth function on B which is extended to a function f̂ = f ∘ π constant along the fibres of π then we define a new metric $g_{f}$ on M by $g_{f} |_{\times} \equiv g_{M} |_{\times}, g_{f} {|_{\times T M ̂} \equiv f ̂ ² g_{M} |}_{\times T M ̂}$ , where and denote the bundles of horizontal and vertical vectors. The manifold $(M, g_{f})$ obtained that way is called a warped submersion. The function f is called a warping function. We show...

The Killing Tensors on an $n$ -dimensional Manifold with $S L (n,)$ -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $S L (n,)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

$η$ -Ricci Solitons on $η$ -Einstein ${(L C S)}_{n}$ -Manifolds

Shyamal Kumar Hui, Debabrata Chakraborty (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The object of the present paper is to study $η$ -Ricci solitons on $η$ -Einstein ${(L C S)}_{n}$ -manifolds. It is shown that if $ξ$ is a recurrent torse forming $η$ -Ricci soliton on an $η$ -Einstein ${(L C S)}_{n}$ -manifold then $ξ$ is (i) concurrent and (ii) Killing vector field.

On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

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Let $(M, g)$ be a compact Riemannian manifold and $P_{g}$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$ . We prove that if $P_{g}$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^{\infty} (M, ℝ)$ for which $P_{e^{f} g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty} (M, ℝ)$ . As a consequence we prove that if $P_{g}$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

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For $n = 2 m \geq 4$ , let $Ω \in ℝ^{n}$ be a bounded smooth domain and $𝒩 \subset ℝ^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that ${u_{k}} \in W^{m, 2} (Ω, 𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$ -polyharmonic maps $\frac{d}{d t} |_{t = 0} E_{m} (Π (u + t ξ)) = 0$ with $Φ_{k} \to 0$ in ${(W^{m, 2} (Ω, 𝒩))}^{*}$ and $u_{k} ⇀ u$ weakly in $W^{m, 2} (Ω, 𝒩)$ . Then $u$ is an $m$ -polyharmonic map. In particular, the space of $m$ -polyharmonic maps is sequentially compact for the weak- $W^{m, 2}$ topology.

On Kakeya–Nikodym averages, $L^{p}$ -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Matthew D. Blair, Christopher D. Sogge (2015)

Journal of the European Mathematical Society

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We extend a result of the second author [27, Theorem 1.1] to dimensions $d \geq 3$ which relates the size of $L^{p}$ -norms of eigenfunctions for $2 < p < 2 (d + 1) / d - 1$ to the amount of $L^{2}$ -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " $ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^{2}$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

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Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , where the sum is taken over the lattice of all points k in $ℝ^{n}$ having integer-valued components, j∈ℕ and $a_{k} \in ℂ$ . Let $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .