### (1,1)-geodesic maps into Grassmann manifolds.

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In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold $(M,g)$ onto a complete Riemannian manifold $(\widehat{M},\widehat{g})$ in terms of developing geodesic triangles of $M$ onto $\widehat{M}$. More precisely, we show that if ${A}_{0}:T{{|}_{{x}_{0}}M\to T|}_{{\widehat{x}}_{0}}\widehat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $\gamma $, $\omega $ of $M$ based at ${x}_{0}$ the development through ${A}_{0}$ of the composite path $\gamma \xb7\omega $ onto $\widehat{M}$ results in a closed path based at ${\widehat{x}}_{0}$, then there exists a Riemannian covering map...

Let $\mathcal{M}=(M,{\mathcal{O}}_{\mathcal{M}})$ be a smooth supermanifold with connection $\nabla $ and Batchelor model ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. From $(\mathcal{M},\nabla )$ we construct a connection on the total space of the vector bundle $E\to M$. This reduction of $\nabla $ is well-defined independently of the isomorphism ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. It erases information, but however it turns out that the natural identification of supercurves in $\mathcal{M}$ (as maps from ${\mathbb{R}}^{1|1}$ to $\mathcal{M}$) with curves in $E$ restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics...

In this note we study the Ledger conditions on the families of flag manifold $({M}^{6}=SU\left(3\right)/SU\left(1\right)\times SU\left(1\right)\times SU\left(1\right),{g}_{({c}_{1},{c}_{2},{c}_{3})})$, $({M}^{12}=Sp\left(3\right)/SU\left(2\right)\times SU\left(2\right)\times SU\left(2\right),{g}_{({c}_{1},{c}_{2},{c}_{3})})$, 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...

We present an algorithm to generate a smooth curve interpolating a set of data on an $n$-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over...

We give necessary and sufficient conditions for Riemannian maps to be biharmonic. We also define pseudo-umbilical Riemannian maps as a generalization of pseudo-umbilical submanifolds and show that such Riemannian maps put some restrictions on the target manifolds.

Y. Euh, J. Park and K. Sekigawa were the first authors who defined the concept of a weakly Einstein Riemannian manifold as a modification of that of an Einstein Riemannian manifold. The defining formula is expressed in terms of the Riemannian scalar invariants of degree two. This concept was inspired by that of a super-Einstein manifold introduced earlier by A. Gray and T. J. Willmore in the context of mean-value theorems in Riemannian geometry. The dimension $4$ is the most interesting case, where...

The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold $(M,g)$ satisfying the first odd Ledger condition is said to be of type $\mathcal{A}$. The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers...

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega}_{\epsilon}$, ${\partial}_{{\nu}_{\epsilon}}u=0$, t > 0, $x\in \partial {\Omega}_{\epsilon}$, $\left({E}_{\epsilon}\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega}_{\epsilon}$ is a thin domain in ${\mathbb{R}}^{l}$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${\mathbb{R}}^{l}$. If G is dissipative, then equation $\left({E}_{\epsilon}\right)$ has a global attractor ${}_{\epsilon}$. We identify a “limit” equation for the family $\left({E}_{\epsilon}\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon}$ as ε → 0⁺.