In this article, we construct some spacelike austere submanifolds in pseduoEuclidean spaces. We also get some indefinite special Lagrangian submanifolds by constructing twisted normal bundle of spacelike austere submanifolds in pseduo-Euclidean spaces.
In this paper, we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into Kiihler manifold. Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V = JMδJM satisfies some decay conditions, we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps. These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.
The Frenet-Serret formula is used to characterize the constant angle ruled surfaces in R3. When the surfaces are the tangent developmental and normal surfaces, that is, r（s, v） = tr（s） ＋v（cosα（s） . t（s） ＋sina（s） . n（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a certain special surface. When the surfaces are normal and binormal surfaces, that is, r （ s, v ） = σ （ s ） ＋ v （ cosa （ s ） . n（s） ＋ since（s） . b（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a cylindrical surface.
The biharmonicity of the product map Φ2=φ×ψ and the two generalized projections φ-and ψ-are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant solution of -1/f2 Jφ（dφ（grad（lnb）））＋n/2 grad｜dφ（grad（lnb））｜2=0 and f is a non-constant solution of -1/b2Jψ（dψ（grad（lnf）））＋m/2grad｜dψ（grad（lnf））｜2=0, and Φ2=φ×ψ is a proper biharmonic map if and only if φ-and ψ-are proper biharmonic maps.
