On the hypersurfaces with non-diagonalizable shape operator in Minkowski spaces

Abstract

Let $M$ be an oriented hypersurface of the Minkowski space $\mathbb E^{n+1}_1$. One of the most important extrinsic object of $M$ is its shape operator $S$ defined by the Wiengarten formula $$SX=-\tilde \nabla_X N,$$ where $N$ is the unit normal vector field to $M$ whenever $X \in TM$. The shape operator can be used to determine how the tangent plane and its normal move in all directions. Note that $S$ is a self adjoint endomorphism in $TM$. Therefore, it is diagonalizable when $M$ is Riemannian. However, if $M$ is Lorentzian, then its shape operator can be non-diagonalizable. In this case, the shape operator $S$ has four canonical forms. These canonical forms are written with respect to either an orthonormal basis or a pseudo-orthonormal basis. If the basis is orthonormal, then it is called a orthonormal frame field. An orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis $\{ E_1, \hdots, E_n \}$ such that $$(E_1,E_1)=-1, \quad (E_1,E_i)=0, \quad (E_i,E_j)=\delta_{ij}$$ for $2 \leq i, \ j \leq n$. On the other hand if the basis is pseudo-orthonormal, then it is called a pseudo-orthonormal frame field. A pseudo-orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis { X, Y, E_1, \hdots, E_{n-2} } such that $$(X,X)= (Y,Y)=0, \quad (X,E_i)=(Y,E_i)=0, \quad (X,Y)=-1$$ and $$(E_i, E_j)=\delta_{ij}$$ for $1 \leq i, \ j \leq n-2$. The eigenvalues and eigenvectors of $S$ are called the principal curvatures and principal directions of $M$, respectively. If the shape operator $S$ is diagonalizable and $M$ has constant principal curvatures, then the hypersurface $M$ is said to be isoparametric. If $S$ is non-diagonalizable and its minimal polynomial is constant, then $M$ is said to be isoparametric. In this thesis, we study isoparametric hypersurfaces with non-diagonalizable shape operator in Minkowski space $\mathbb E^{4}_1$. This thesis consists of five sections. First section is introduction. In the second section, we give some basic concepts on Lorentzian inner product and also some basic facts on hypersurfaces of $\mathbb E^{n+1}_1$. In the third section, a theorem about isoparametric hypersurfaces is given. We note that these theorems are proved by Magid in 1985. We prove these theorems by using another method. In fact, this theorem implies that there is only four classes of isoparametric hypersurface using the Codazzi and Gauss equations in $\mathbb E^{4}_1$. Then, we give another theorem which proves that there is no isoparametric hypersurface in $\mathbb E^{4}_1$ with complex principal curvatures. In the fourth section, we construct a family of hypersurfaces with non-diagonalizable shape operator in $\mathbb E^{5}_1$. We obtain the shape operator, the mean curvature, Gauss-Kronecker curvature and Levi-Civita connection of this hypersurface. Then, we give the necessary and sufficient condition for this hypersurface to be minimal with a theorem.