Journal of Geometry

Submanifolds of Em with pointwise planar normal sections were studied in [1] and others. In the present paper, we will prove that an isotropic submanifold in Em with pointwise planar normal sections is isometric to a symmetric space of rank one or to a Euclidean space. Moreover we will determine such surfaces in Em with the above assumptions.

Pick’s theorem relates the number of lattice points to the area for a lattice polygon. Diaz and Robins gave a proof of Pick’s theorem by using the Weierstrass $$\wp $$ -function and complex analysis. As an analogue to lattice convex polyhedra, Reeve’s theorem is known as a solid version of Pick’s theorem. In this paper, we study counting lattice points on a lattice polyhedron by using vector analysis, and we extend Reeve’s theorem to nonconvex polyhedral complexes.

A new angular measure in Minkowski plane was introduced recently. It is additive, invariant under invertible linear transformations, and has a clear interpretation as an "amount of rotation" from one direction to another one. The total angle τ around a point depends on the unit ball. It is known that $$4.443 \approx \sqrt{2}\pi \leq \tau \leq 8$$ . We show that $$4.985 \approx 4 \sqrt{2}\, {\rm ln}\,{\rm tan} \frac{3\pi}{8} \leq \tau \leq 8\, {\rm ln}\,{\rm tan} \frac{3\pi}{8}\, \approx 7.051$$ .

We prove two comparison theorems for the volume of a geodesic ball in a Riemannian manifold taking as a model a geodesic ball in a product of two space forms.

The Suetake planes and the simple full non-square André planes are shown to be exactly those finite translation planes that are coupled with a semifield plane. Also, the Suetake planes are never generalized André planes.

In this paper we are concerned with a special kind of subsets of finite projective planes and give some new examples of totally m-regular (k,n)-arcs.