Mirjana Djorić

Three-Dimensional Lorentz Metrics and Curvature Homogeneity of Order One

In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k ∈ N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.

CR submanifolds of complex projective space

1. Complex manifold.- 2. Almost complex structure.- 3. Complex vector space complexification.- 4. Kahler manifold.- 5. Structure equations of a submanifold.- 6. Submanifolds of a Euclidean space.- 7. Submanifolds of a complex manifold.- 8. The Levi form.- 9. The principal circle bundle S^{2n+1}({\bf P}^n({\bf C}),S^1).- 10. Submersion and immersion.- 11. Hypersurfaces of a Riemannian manifold of constant curvature.- 12. Hypersurfaces of a sphere S^{n+1}(1/a).- 13. Hypersurfaces of a sphere with parallel shape operator.- 14. Codimension reduction of a submanifold.- 15. CR submanifolds of maximal CR dimension.- 16. Real hypersurfaces of a complex projective space.- 17. Tubes around submanifolds.- 18. Levi form of CR submanifolds of maximal CR dimension of a complex space form.- 19. Eigenvalues of the shape operator A of CR submanifolds of maximal CR dimension of a complex space form.- 20. CR submanifolds of maximal CR dimension satisfying the condition h(FX,Y)+h(X,FY)=0.- 21. Contact CR submanifolds of maximal CR dimension.- 22. Invariant submanifolds of real hypersurfaces of complex space forms.- 23. The scalar curvature of CR submanifolds of maximal CR dimension.

THREE-DIMENSIONAL CR-SUBMANIFOLDS IN THE NEARLY KAHLER 6-SPHERE WITH ONE-DIMENSIONAL NULLITY

In this paper, we study certain three-dimensional CR-submanifolds M of the nearly Kahler 6-dimensional sphere S6(1). It is well known that there does not exist a three-dimensional totally geodesic proper CR-submanifold in S6(1). In this paper we obtain a classification of the 3-dimensional CR-submanifolds which are the closest possible to totally geodesic submanifolds, i.e. those that admit a one-dimensional nullity distribution.

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.

Geometric Structures as Determined by the Volume of Generalized Geodesic Balls

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Geometry of tubes about characteristic curves on Sasakian manifolds

(M, {\cal G})

A theorem of Archimedes about spheres and cylinders and two-point homogeneous spaces

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CR submanifolds of maximal CR dimension

(M, D{\cal G})

Levi Form of CR Submanifolds of Maximal CR Dimension of Complex Space forms

, where D is a torsion-free and Ricci-symmetric connection, appears in several geometric situations. We study the Taylor expansion in this case, where all metric notions are Riemannian, while now the exponential mapping is induced from the connection D. We give many applications, in particular in different hypersurface theories.

Geodesic lines on nearly Kähler S3×S3

One studies, using Riemannian foliation theory, some aspects of the intrinsic and extrinsic geometry of small tubes about the flow lines of the characteristic vector field on a Sasakian manifold. In particular, one focuses on some characteristic properties of the shape operator and the Ricci operator of these tubes for the classes of ϕ-symmetric spaces and Sasakian space forms.