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## Notes on projective differential geometry

Eastwood, Michael George
Tipo: Parte de Livro
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Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ‘Bernstein-Gelfand-Gelfand machinery’. As such, it is completely parallel to conformal differential geometry. On the other hand, there are direct applications within Riemannian differential geometry. We shall soon see, for example, a good geometric reason why the symmetries of the Riemann curvature tensor constitute an irreducible representation of SL(n,ℝ) (rather than SO(n) as one might naively expect). Projective differential geometry also provides the simplest setting in which overdetermined systems of partial differential equations naturally arise.; http://www.springer.com/math/dyn.+systems/book/978-0-387-73830-7; Michael Eastwood

## Extended Derdzinski-Shen theorem for the Riemann tensor

Mantica, Carlo Alberto; Molinari, Luca Guido
Tipo: Artigo de Revista Científica
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We extend a classical result by Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of some well known differential structures.; Comment: 5 pages

## Conformally Osserman manifolds

Nikolayevsky, Yuri
Tipo: Artigo de Revista Científica
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An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne 3, 4, 16$ is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space.; Comment: 23 pages

## Weyl compatible tensors

Mantica, Carlo A.; Molinari, Luca G.
Tipo: Artigo de Revista Científica
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We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular it is shown that the existence of a Weyl compatible vector implies the Weyl tensor to be algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen, Hall) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as McIntosh et al.). Hypersurfaces of pseudo Euclidean spaces provide a simple example of Weyl compatible Ricci tensor.; Comment: 14 pages. References and comments added

## Geometric Realizations of Hermitian curvature models

Brozos-Vazquez, M.; Gilkey, P.; Kang, H.; Nikcevic, S.
Tipo: Artigo de Revista Científica
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We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question.

## Geometry of CR submanifolds of maximal CR dimension in complex space forms

Milijevic, Mirjana
Tipo: Artigo de Revista Científica
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On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.; Comment: 8 pages

## Non-existence of CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms

Milijevic, Mirjana
Tipo: Artigo de Revista Científica
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It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.; Comment: 8 pages

## R-separation of variables for the conformally invariant Laplace equation

Chanachowicz, M.; Chanu, C.; McLenaghan, R. G.
Tipo: Artigo de Revista Científica
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The conditions for R-separation of variables for the conformally invariant Laplace equation on an n-dimensional Riemannian manifold are determined and compared with the conditions for the additive separation of the null geodesic Hamilton-Jacobi equation. The case of 3-dimensions is examined in detail and it is proven that on any conformally flat manifold the two equations separate in the same coordinates.; Comment: 13 pages, submitted to Journal of Geometry and Physics. Replaced due to a factor of 1/4 error found in some presented formulae; note this does not affect the results derived and discussed - only in the initial presentation

## Group-theoretic Description of Riemannian Spaces

Samokhvalov, Serhiy E.
Tipo: Artigo de Revista Científica
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It is shown that a locally geometrical structure of arbitrarily curved Riemannian space is defined by a deformed group of its diffeomorphisms; Comment: 14 pages

## Canonical Deformed Groups of Diffeomorphisms and Finite Parallel Transports in Riemannian Spaces

Samokhvalov, Serhiy E.
Tipo: Artigo de Revista Científica
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We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are equivalent.; Comment: 12 pages

## Geometric Realizations of para-Hermitian curvature models

Brozos-Vazquez, M.; Gilkey, P.; Nikcevic, S.; Vazquez-Lorenzo, R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri-Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen to have constant scalar curvature and constant *-scalar curvature.

## Transforms for minimal surfaces in the 5-sphere

Bolton, J.; Vrancken, L.
Tipo: Artigo de Revista Científica
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We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space. We illustrate this explicitly in the case of bipolar minimal surfaces.

## Powers of the space forms curvature operator and geodesics of the tangent bundle

Saharova, Yelena; Yampolsky, Alexander
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space form and give a unified proof of the following property: all geodesic curvatures of projected curve are zero starting from k_3,k_6 and k_{10} for the real, complex and quaternionic space formes respectively.; Comment: 14 pages

## Inhomogeneous Ambient Metrics

Graham, C. Robin; Hirachi, Kengo
Tipo: Artigo de Revista Científica
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An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedure for taking the smooth part of such an inhomogeneous ambient metric. The metrics which result depend on the choice of an "ambiguity tensor" as well as a conformal class. An application to the description of scalar conformal invariants in even dimensions is outlined.; Comment: 18 pages

## Regularization by sup-inf convolutions on Riemannian manifolds: an extension of Lasry-Lions theorem to manifolds of bounded curvature

Azagra, Daniel; Ferrera, Juan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of bounded sectional curvature. More specifically, among other things we show that if the sectional curvature $K$ of $M$ satisfies $-K_0\leq K\leq K_0$ on $M$ for some $K_0>0$, and if the injectivity and convexity radii of $M$ are strictly positive, then every bounded, uniformly continuous function $f:M\to\mathbb{R}$ can be uniformly approximated by globally $C^{1,1}$ functions defined by $$(f_{\lambda})^{\mu}=\sup_{z\in M}\inf_{y\in M}\{f(y)+\frac{1}{2\lambda} d(z,y)^{2}-\frac{1}{2\mu}d(x,z)^2\}$$ as $\lambda, \mu\to 0^{+}$, with $0<\mu<\lambda/2$. Our definition of (global) $C^{1,1}$ smoothness is intrinsic and natural, and it reduces to the usual one in flat spaces, but we warn the reader that, in the noncompact case, this definition differs from other notions of (rather local) $C^{1,1}$ smoothness that have been recently used, for instance, by A. Fathi and P. Bernard (based on charts). The importance of this regularization method lies (rather than on the degree of smoothness obtained) on the fact that the correspondence $f\mapsto (f_{\lambda})^{\mu}$ is explicit and preserves many significant geometrical properties that the given functions $f$ may have...

## Remarks on Bianchi sums and Pontrjagin classes

Labbi, Mohammed Larbi
Tipo: Artigo de Revista Científica
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We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler-Poincar\'e charateristic and the $k$-th Pontrjagin number of a $4k$-dimensional Thorpe manifold.; Comment: 19 pages

## Interactions between the composition and exterior products of double forms and applications

Belkhirat, A.; Labbi, M. L.
Tipo: Artigo de Revista Científica
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We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms.\\ We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its Pontrjagin class of degree $4k$ vanishes.; Comment: 24 pages

## Levi-Civita's Theorem for Noncommutative Tori

Rosenberg, Jonathan
Tipo: Artigo de Revista Científica
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We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.; Comment: Proposition 3.4 corrected

## Weakly Z symmetric manifolds

Mantica, Carlo A.; Molinari, Luca G.
Tipo: Artigo de Revista Científica