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Abstract.  In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition. [ABSTRACT FROM AUTHOR]

We study minimal Lagrangian immersions from an indefinite real space form $M^n_s(c)$ into an indefinite complex space form ${\tilde{\mathbb{M}}}^n_s(4\tilde c)$. Provided that $c \ne \tilde c$, we show that $M^n_s(c)$ has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that $c = \tilde c$, this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal). [ABSTRACT FROM AUTHOR]

We study affine hyperspheres $M$ with constant sectional curvature (with respect to the affine metric $h$). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either \[(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2}) = 1] or \[(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2})x_{2m+1} = 1] where the dimension $n$ satisfies $n=2m-1$ or $n=2m$. Up to now, this conjecture was proved if $M$ is positive definite or if $M$ is a $3$-dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres. [ABSTRACT FROM AUTHOR]

In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb{C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb{C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen's equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen's equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$. [ABSTRACT FROM AUTHOR]

For (positive) definite surfaces in $\mathbb{R}^{4}$ there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection $\nabla $. We classify all surfaces in $\mathbb{R}^{4}$ with planar $\nabla $-geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle. [ABSTRACT FROM AUTHOR]