Related papers: Compatibility of Gauss maps with metrics

We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm's method, Riemannian submersions and connected sums.

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form…

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $\Lambda$ is its DN-map, and $\mathscr E:M\to\mathbb{C}^{n}$ % $\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\partial M=\partial M'$; let…

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a…

Let L be an algebraic set and let g : R^(n+1) \times L --> R^(2n) (n is even) be a polynomial mapping such that for each l in L there is r(l)>0 such that the mapping g_l = g(.,l) restricted to the sphere S^n(r) is an immersion for every…

We prove that the metric of the Riemannian product $(\mb{S}^k(r_1)\times \mb{S}^{n-k}(r_2), g^n_k)$, $r_1^2+r_2^2=1$, is a Yamabe metric in its conformal class if, and only if, either $g^n_k$ is Einstein, or the linear isometric embedding…

We establish the lower bound of $4\pi(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\mathbb{R}^4$. This lower bound is…

As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen…

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding…

On the Grassmann manifold G (m, n) of m-dimensional subspaces of an n-dimensional projective space P^n, a certain supplementary construction called the normalization is considered. By means of this normalization, one can construct the…

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge of the…

We $\delta$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_\delta:X^m\to Y^N$ with curvatures $curv(f_\delta) < \frac {\sqrt…

Let $X$ be a compact real algebraic set of dimension $n$. We prove that every Euclidean continuous map from $X$ into the unit $n$-sphere can be approximated by regulous map. This strengthens and generalizes previously known results.

Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian…

In this paper, we study the gluing construction of the extended harmonic maps between Riemannian manifolds. Harmonic maps are critical points of the energy functional. We construct the gluing map of the extended harmonic maps from Riemann…

The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq…

We consider two classes of smooth maps M^n\to C ^N. Definition. A map f:M^n\to C^N is called an independent map if df_1(p)\wedge...\wedge df_N (p)\neq 0. We are interested in the optimal value of N for all manifolds of dimension n for…

We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is…