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unique real number ϕ ( X, Y ) between 0 and π such that 

the Lorentzian spacelike angle between X and Y is defined to be ϕ ( X, Y ) [8]. 

Let X and Y be spacelike vectors in R 3 1 that span a timelike vector subspace. Then we have 

Hence, there is a unique positive real number ϕ ( X, Y ) between 0 and π such that 

the Lorentzian timelike angle between X and Y is defined to be ϕ ( X, Y ) [8]. 

Let X be a spacelike vector and Y be a positive timelike vector in R 3 $_{1}$. Then there is a unique nonnegative real number ϕ ( X, Y ) such that 

the Lorentzian timelike angle between X and Y is defined to be ϕ ( X, Y ) [8]. 

Let M be a semi-Riemannian submanifold of M and D be the Levi-Civita connection of M, the function II : χ ( M ) × χ ( M ) −→ χ ( M $_{)}$⊥ such that 

is ℑ ( M ) − bilinear and symmetric. II is callled the shape tensor (or second fundamental form tensor) of M ⊂ M [5]. 

Let N be a unit normal vector field on a semi-Riemannian hypersurface 

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