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We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique…

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits…

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and…

In this paper we study a geometric configuration of submanifolds of arbitrary codimension in an ambient Riemannian space. We obtain relations between the geometry of a q-codimension submanifold Mn along its boundary and the geometry of the…

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We…

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean…

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly…

Given an embedded stable hypersurface in a four-dimensional symplectic manifold, we prove that it is stable isotopic to a $C^0$-close stable hypersurface with the following property: $C^\infty$-nearby hypersurfaces are generically unstable.…

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "\`a la Margulis" for the systole and a volume estimate for…

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2…

We prove that the upper metric mean dimension of $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, coincides with the dimension of the manifold. In the case of continuous…

We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the…

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a…

We show that in an $8$-dimensional closed Riemmanian manifold with $C^\infty$-generic metrics, every minimal hypersurface is smooth and nondegenerate. This confirms a full generic regularity conjecture of minimal hypersurfaces in dimension…

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds…

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics…

We establish generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight. In particular, we show that every isoperimetric region has a smooth nondegenerate boundary for a generic choice of smooth…

We investigate suitable, physically motivated conditions on spacetimes containing certain submanifolds - the so-called {weakly trapped submanifolds} - that ensure, in a set of neighboring metrics with respect to a convenient topology, that…

In this paper we investigate possible extensions of the idea of geodesic completeness in complex manifolds, following two directions: metrics are somewhere allowed not to be of maximum rank, or to have 'poles' somewhere else. Geodesics are…

It is a question by C.Sormani that whether there exists a $k \in \mathbb N$, such that any compact, smooth and simply connected manifold has a 1/k-geodesic. We prove in this paper that this is not true by showing for each $k$, there exists…