Riemannian submersions and the regular interval theorem of Morse theory

A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemannian submersions (for real valued functions) and Morse theory.Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,) R be a smooth real valued function on a non-compact complete connected Riemannian manifold (M,g) such that df is bounded in norm away from zero. By pointwise conformally deforming g to pg, p = d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]², we show that (M,pg) is a complete Riemannian manifold, and that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,pg) R is a surjective Riemannian submersion and a globally trivial fiber bundle over R. In particular, all of the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] are diffeomorphic, and M is globally diffeomorphic to the product bundle R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] ^–1(0) by a diffeomorphism F ⁰: R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) M that straightens out the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\].Moreover, we show that (F ⁰)^*(pg) is a parameterized Riemannian product manifold on R×% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), i.e., a product manifold with a metric that varies on the fibers {t} × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0). Also, F ⁰: (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0),(F ⁰)^*(pg)) (M,g) is a conformal diffeomorphism between the Reimannian manifolds (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), (F ⁰)^*(pg)) and (M,g),so that (M,g) is conformally equivalent to a parameterized Riemannian product manifold. The conformal diffeomorphism F ⁰ is an isometry between the Riemannian product manifold (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), 1 + g ₀) (where g ₀) is the metric induced by g on % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) and (M,g) if and only if d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 1 and Hess % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 0.     

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

     

Vector versions of a density theorem and applications to problems of control theory

A vector version of a density theorem related to problems of control theory and extending results of Warga, Ghouila-Houri, Castaing, and Berliocchi-Lasry is presented. Next, some of its applications to control problems with operator constraints are discussed. A result concerning existence of classical control for a relaxed Pareto variational problem with contraints of this type is also given.     

A theorem in fine potential theory and applications to finely holomorphic functions

Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

A generalization of a theorem of Imai and its applications to Iwasawa theory

It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite $K(K^{1/p^\infty })$ -rational torsion subgroup. This can be used to remove certain conditions assumed in some theorems in Iwasawa theory.     

The space of strong types and an open map theorem

We define a new topology on the space of strong types on a subsetA of a model of a given theory and prove that either . We also deduce an open map theorem.     

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f _n } ∪L _p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ _n a _n T ⁿ g/ n whenT is anL ₂ -contraction,g∃L ₂ , and {a _n } is an appropriate sequence. Given a sequence {f _n }∪L _p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on andg∈L ₂ (π), the random power series Σ _n f _n (x)T ⁿ g converges π-a.e. The conditions are used to show that for {f _n } centered i.i.d. withf ₁∃L log⁺ L, there exists a set ofx of full measure such that for every contractionT on andg∃L ₂ (π), the random series Σ _n f _n (x)T ⁿ g/n converges π-a.e. We use Menchoff's own spelling of his name in the papers he wrote in French. Dedicated to Hillel Furstenberg upon his retirement     

Ingham-type inequalities for complex frequencies and applications to control theory

For complex valued sequences of the form ωn=an+ibn with an∈R and bn?0, we prove inequalities of the form , for all sequences {xn} with . We apply these to prove exact null-controllability for a class of hinged beam equations with mild internal damping with either boundary control or internal control.     

Fixed points for generalized contractions and applications to control theory

The concepts of “weak/strong topological contraction” and a generalization of Banach contraction mappings called “p$p$-contraction” are introduced and used to prove fixed point theorems for self-mappings from a topological/metric space into itself satisfying topological contraction/metric p$p$-contraction, respectively. Certain non-linear integral equations defined on C[a,b]$C[a,b]$ satisfying generalized Lipschitzian conditions can easily be solved by applying these theorems. In the sequel, we shall study the possibility of optimally controlling the solution of the ordinary differential equation via dynamic programming.     

A simplified proof of arithmetical completeness theorem for provability logic GLP

We present a simplified proof of Japaridze’s arithmetical completeness theorem for the well-known polymodal provability logic GLP. The simplification is achieved by employing a fragment J of GLP that enjoys a more convenient Kripke-style semantics than the logic considered in the papers by Ignatiev and Boolos. In particular, this allows us to simplify the arithmetical fixed point construction and to bring it closer to the standard construction due to Solovay.     

A viability approach to the inverse set-valued map theorem

The purpose of this paper is to characterize by means of viability tools the pseudo-lipschitzianity property of a set-valued map F in a neighborhood of a point of its graph in terms of derivatives of this set-valued map F in a neighborhood of a point of its graph, instead of using the transposes of the derivatives. On the way, we relate these properties to the calmness index of a set-valued map, an extensions of Clarke’s calmness of a function, as well as Doyen’s Lipschitz kernel of a set-valued map, which is the largest Lipschitz submap.