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 $M^{12}$

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

Really Natural Linear Indexed Type Checking

Recent works have shown the power of linear indexed type systems for enforcing complex program properties. These systems combine linear types with a language of type-level indices, allowing more fine-grained analyses. Such systems have been fruitfully applied in diverse domains, including implicit complexity and differential privacy. A natural way to enhance the expressiveness of this approach is by allowing the indices to depend on runtime information, in the spirit of dependent types. This approach is used in DFuzz, a language for differential privacy. The DFuzz type system relies on an index language supporting real and natural number arithmetic over constants and variables. Moreover, DFuzz uses a subtyping mechanism to make types more flexible. By themselves, linearity, dependency, and subtyping each require delicate handling when performing type checking or type inference; their combination increases this challenge substantially, as the features can interact in non-trivial ways. In this paper, we study the type-checking problem for DFuzz. We show how we can reduce type checking for (a simple extension of) DFuzz to constraint solving over a first-order theory of naturals and real numbers which, although undecidable, can often be handled in practice by standard numeric solvers

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