Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants $|R|^2$ and $|Ric|^2$ of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions.Comment: 18 pages, LaTeX. Added a few lines in the introduction, corrected a few typos. Final version. Accepted for publication in GAF

Constant Jacobi osculating rank of U(3)/(U(1) x U(1) x U(1)) -- Appendix--

Este es el apéndice del documento [T. Arias-Marco, Constant Jacobi osculating rank of U(3)/(U(1) × U(1) × U(1)), Arch. Math. (Brno) 45 (2009), 241–254]This is the appendix of the paper [T. Arias-Marco, Constant Jacobi osculating rank of U(3)/(U(1) × U(1) × U(1)), Arch. Math. (Brno) 45 (2009), 241–254] where we obtain an interesting relation between the co-variant derivatives of the Jacobi operator valid for all geodesic on the flag manifold M6 = U(3)/(U(1)×U(1)×U(1)). As a consequence, an explicit expression of the Jacobi operator independent of the geodesic can be obtained on such a manifold. Moreover, we show the way to calculate the Jacobi vector fields on this manifold by a new formula valid on every g.o. space.Trabajo patrocinado por la Dirección General de Investigación (España)y el Proyecto FEDER MTM 2007-65852,la red MTM2008-01013-E/ y por DFG Sonderforschungsbereich 647

A property of Wallach's flag manifolds

summary:In this note we study the Ledger conditions on the families of flag manifold $(M^{6}=SU(3)/SU(1)\times SU(1) \times SU(1), g_{(c_1,c_2,c_3)})$, $\big (M^{12}=Sp(3)/SU(2) \times SU(2) \times SU(2), g_{(c_1,c_2,c_3)}\big )$, constructed by N. R. Wallach in (Wallach, N. R., Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293.). In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D’Atri space if and only if it is naturally reductive. Therefore, we finish the study of $M^6$ made by D’Atri and Nickerson in (D’Atri, J. E., Nickerson, H. K., Geodesic symmetries in spaces with special curvature tensors, J. Differenatial Geom. 9 (1974), 251–262.). Moreover, we correct and improve the result given by the author and A. M. Naveira in (Arias-Marco, T., Naveira, A. M., A note on a family of reductive Riemannian homogeneous spaces whose geodesic symmetries fail to be divergence-preserving, Proceedings of the XI Fall Workshop on Geometry and Physics. Publicaciones de la RSME 6 (2004), 35–45.) about $M^{12}$

Geodesic graphs for geodesic orbit Finsler (α,β)(\alpha,\beta) metrics on spheres

Invariant geodesic orbit Finsler $(\alpha,\beta)$ metrics $F$ which arise from Riemannian geodesic orbit metrics $\alpha$ on spheres are determined. The relation of Riemannian geodesic graphs with Finslerian geodesic graphs proved in a previous work is now illustrated with explicit constructions. Interesting examples are found such that $(G/H,\alpha)$ is Riemannian geodesic orbit space, but for the geodesic orbit property of $(G/H,F)$ the isometry group has to be extended. It is also shown that projective spaces other than ${\mathbb{R}}P^n$ do not admit invariant purely Finsler $(\alpha,\beta)$ metrics

Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica.

Nowadays, the concept of homogeneity is one of the fundamental notions in geometry although its meaning must be always specified for the concrete situations. In this thesis, we consider the homogeneity of Riemannian manifolds and the homogeneity of manifolds equipped with affine connections. The first kind of homogeneity means that, for every smooth Riemannian manifold (M, g), its group of isometries I(M) is acting transitively on M. Part I of this thesis fits into this philosophy. Afterwards in Part II, we treat the homogeneity concept of affine connections. This homogeneity means that, for every two points of a manifold, there is an affine diffeomorphism which sends one point into another. In particular, we consider a local version of the homogeneity, that is, we accept that the affine diffeomorphisms are given only locally, i.e., from a neighborhood onto a neighborhood. More specifically, we devote the first Chapter of Part I to make a brief overview of some special kinds of homogeneous Riemannian manifolds which will be of special relevance in Part I and to show how the software MATHEMATICA© becomes useful. For that, we prove that the three-parameter families of flag manifolds constructed by N. R. Wallach in "Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), p. 276293" are DAtri spaces if and only if they are naturally reductive spaces. Thus, we improve the previous results given by DAtri, Nickerson and by Arias-Marco, Naveira. Moreover, in Chapter 2 we obtain the complete 4-dimensional classification of homogeneous spaces of type A. This allows us to prove correctly that every 4-dimensional homogeneous DAtri space is naturally reductive. Therefore, we correct, complete and improve the results presented by Podestà, Spiro, Bueken and Vanhecke. Chapter 3 is devoted to prove that the curvature operator has constant osculating rank over g.o. spaces. It is mean that a real number r exists such that under some assumptions, the higher order derivatives of the curvature operator from 1 to r are linear independent and from 1 to r + 1 are linear dependent. As a consequence, we also present a method valid on every g.o. space to solve the Jacobi equation. This method extends the method given by Naveira and Tarrío for naturally reductive spaces. In particular, we prove that the Jacobi operator on Kaplans example (the first known g.o. space that it is not naturally reductive) has constant osculating rank 4. Moreover, we solve the Jacobi equation along a geodesic on Kaplans example using the new method and the well-known method used by Chavel, Ziller and Berndt,Tricerri, Vanhecke. Therefore, we are able to present the main differences between both methods. In Part II, we classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Therefore, we generalize the result given by Opozda for torsion-less case. Moreover, from our computations we obtain interesting consequences as the relation between the classifications given for the torsion less-case by Kowalski, Opozda and Vláek. In addition, we obtain interesting consequences about flat connections with torsion. In general, the study of these problems sometimes requires a big number of straightforward symbolic computations. In such cases, it is a quite difficult task and a lot of time consuming, try to make correctly this kind of computations by hand. Thus, we try to organize our computations in (possibly) most systematic way so that the whole procedure is not excessively long. Also, because these topics are an ideal subject for a computer-aided research, we are using the software MATHEMATICA©, throughout this work. But we put stress on the full transparency of this procedure. __________________________________________________________________________________________________ RESUMEN En esta tesis, se consideran dos tipos bien diferenciados de homogeneidad: la de las variedades riemannianas y la de las variedades afines. El primer tipo de homogeneidad se define como aquel que tiene la propiedad de que el grupo de isometrías actúa transitivamente sobre la variedad. La Parte I, recoge todos los resultados que hemos obtenido en esta dirección. Sin embargo, en la Parte II se presentan los resultados obtenidos sobre conexiones afines homogéneas. Una conexión afín se dice homogénea si para cada par de puntos de la variedad existe un difeomorfismo afín que envía un punto en otro. En este caso, se considera una versión local de homogeneidad. Más específicamente, la Parte I de esta tesis está dedicada a probar que "las familias 3-paramétricas de variedades bandera construidas por Wallach son espacios de D'Atri si y sólo si son espacios naturalmente reductivos". Más aún, en el segundo Capítulo, se obtiene la clasificación completa de los espacios homogéneos de tipo A cuatro dimensionales que permite probar correctamente que todo espacio de DAtri homogéneo de dimensión cuatro es naturalmente reductivo. Finalmente, en el tercer Capítulo se prueba que en cualquier g.o. espacio el operador curvatura tiene rango osculador constante y, como consecuencia, se presenta un método para resolver la ecuación de Jacobi sobre cualquier g.o. espacio. La Parte II se destina a clasificar (localmente) todas las conexiones afines localmente homogéneas con torsión arbitraria sobre variedades 2-dimensionales. Para finalizar el cuarto Capítulo, se prueban algunos resultados interesantes sobre conexiones llanas con torsión. En general, el estudio de estos problemas requiere a veces, un gran número de cálculos simbólicos aunque sencillos. En dichas ocasiones, realizarlos correctamente a mano es una tarea ardua que requiere mucho tiempo. Por ello, se intenta organizar todos estos cálculos de la manera más sistemática posible de forma que el procedimiento no resulte excesivamente largo. Este tipo de investigación es ideal para utilizar la ayuda del ordenador; así, cuando resulta conveniente, utilizamos la ayuda del software MATHEMATICA para desarrollar con total transparencia el método de resolución que más se adecua a cada uno de los problemas a resolver

Espacios s-simétricos y espacios naturalmente reductivos en dimensiones bajas

El objetivo de este escrito es facilitar al lector interesado en el estudio de los espacios homogéneos naturalmente reductivos y los espacios s-simétricos, el acceso a la historia de los mismos, a los conocimientos previos necesarios para su comprensión, a los últimos avances realizados en este tipo de espacios Riemannianos y a diversos problemas abiertos. Para ello, el primer capítulo de este libro está dirigido a unificar las definiciones y clarificar los resultados sobre los definitivamente denominados espacios s – simétricos, y en el segundo capítulo se estudia el problema de la clasificación de los mismos. Los dos últimos capítulos del libro están dedicados al estudio de los espacios naturalmente reductivos y al problema de su clasificación.The objective of this paper is to provide the reader interested in the study of naturally reductive homogeneous spaces and the s-symmetric spaces, the access to the history of the same, to the previous knowledge necessary for their understanding, to the recent progress in this type of Riemannian spaces and various open problems. To do this, the first chapter of this book is intended to unify the definitions and clarify the results on the definitively called s – symmetrical spaces, and in the second chapter examines the problem of the classification of the same. The last two chapters of the book are dedicated to the study of naturally reductive spaces and the problem of their classification

D'atri spaces of type k and related classes of geometries concerning jacobi operators

In this article we continue the study of the geometry of $k$-D'Atri spaces, $$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown that ${\operatorname{tr}}R_{v}$, ${\operatorname{tr}}R_{v}^{2}$ are invariant under the geodesic flow for any unit tangent vector $v$. Here, assuming that the Riemannian manifold is a D'Atri space, we prove in our main result that ${\operatorname{tr}}R_{v}^{3}$ is also invariant under the geodesic flow if $k\geq 3$. In addition, other properties of Jacobi operators related to the Ledger conditions are obtained and they are used to give applications to Iwasawa type spaces. In the class of D'Atri spaces of Iwasawa type, we show two different characterizations of the symmetric spaces of noncompact type: they are exactly the $\frak{C}$-spaces and on the other hand they are $k$ -D'Atri spaces for some $k\geq 3.$ In the last case, they are $k$-D'Atri for all $k=1,...,n-1$ as well. In particular, Damek-Ricci spaces that are $k$-D'Atri for some $k\geq 3$ are symmetric. Finally, we characterize $k$-D'Atri spaces for all $k=1,...,n-1$ as the $\frak{SC}$-spaces (geodesic symmetries preserve the principal curvatures of small geodesic spheres). Moreover, applying this result in the case of 4% -dimensional homogeneous spaces we prove that the properties of being a D'Atri (1-D'Atri) space, or a 3-D'Atri space, are equivalent to the property of being a $k$-D'Atri space for all $k=1,2,3$.Comment: 19 pages. This paper substitute the previous one where one Theorem has been deleted and one section has been adde