Our aim in this paper is to focus on some applications in differential geometry of the metallic means family (a generalization of the golden mean) and generalized Fibonacci sequences, using a class of polynomial structures defined on Riemannian manifolds. We search for properties of the induced structure on a submanifold by metallic Riemannian structures and we find a necessary and sufficient c...

For an n-dimensional complete connected Riemannian manifold M with sectional curvature KM ≥ 1 and diameter diam(M) > π2 , and a closed connected totally geodesic submanifold N of M , if there exist points x ∈ N and y ∈ M satisfying the distance d(x, y) > π 2 , then N is homeomorphic to a sphere. We also give a counterexample in 2-dimensional case to the following problem: let M be an ndimension...

We study a germ of real analytic n-dimensional submanifold of C that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisor...

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which characterized by the condition that second form proportional to metric. It also generalization of notion geodesic submanifold. In this paper, we classify congruent classes full submanifolds non-flat space forms, and consider their moduli spaces. As consequence, show some spaces isometric immersions bet...

Given N a non generic smooth CR submanifold of C, N = {(N, h(N))} where N is generic in C and h is a CR map from N into C. We prove, using only elementary tools, that if h is decomposable at p′ ∈ N then any decomposable CR distribution on N at p = (p′, h(p′)) extends holomorphically to a complex transversal wedge. This gives an elementary proof of the well known equivalent for totally real non ...

Let M be an n-dimensional submanifold in the simply connected space form F(c) with c + H > 0, where H is the mean curvature of M . We verify that if M(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies RicM ≥ (n−2)(c+H2), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n+1)-sphere with n = even, or CP ( 4 3 (c+H )) ...

For 1 < p < oo we show that LP functions defined on a C°° totally real submanifold of C" can be locally decomposed into the sum of boundary values of holomorphic functions in wedges such that the boundary values are in LP . The general case of a C°° totally real submanifold is reduced to the flat case of R" in C" by an almost analytic change of variables. LP results in the flat case are then ob...

‎Based on a definition for circle in Finsler space‎, ‎recently proposed by one of the present authors and Z‎. ‎Shen‎, ‎a natural definition of extrinsic sphere in Finsler geometry is given and it is shown that a connected submanifold of a Finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field‎, ‎if and only if its circles coincide with circles of the ambient...

In this paper, we establish the following result: Let M be an n-dimensional complete totally real minimal submanifold immersed in CPn with Ricci curvature bounded from below. Then either M is totally geodesic or infr ≤ (3n+1)(n−2)/3, where r is the scalar curvature of M .