Let D be a set of smooth vector fields on the smooth manifold M .It is known that orbits of D are submanifolds of M. Partition F of M into orbits of D is a singular foliation. In this paper we are studying geometry of foliation which is generated by orbits of a family of Killing vector fields.In the case M = R 3 it is obtained full geometrical classification of F . Throughout this paper the word "smooth" refers to a class C ∞ .Let M is a smooth, connected Riemannian manifold of dimension n, X is a smooth vector field, X t (x) is an integral curve passing through x for t = 0. Definition 1. Vector field X on M is called a Killing field if the infinitesimal transformation x → X t (x) is an isometry of M for any t.Example 1. In the three-dimensional Euclidean space R 3 (x, y, z), there are six linearly independent Killing fields on the field of real numbers:The groups of transformations generated by vector fields X 1 , X 2 , X 3 are groups of translations in the direction of the axes Ox, Oy and Oz, respectively, and the groups of transformations generated by last three vector fields are rotations around the axes of the Ox, Oy and Oz accordingly.The last three fields are Killing fields on the sphere S 2 too. Example 2. Consider the three-dimensional sphere S 3 in R 4 ≈ C 2 with the induced metric. Let (x 1 , x 2 , x 3 , x 4 ) is a point on the sphere S 3 . With complex numbers S 3 can be described as follows:Consider in R 4 Killing vector fieldIt is easy to check that this vector field is tangent to the sphere. For a point (z 1 , z 2 ) inS 3 the integral curve of the vector field X, starting from the point (z 1 , z 2 ) for t = 0 has the formIt is obvious that the integral curve γ(t) is circle. The family of integral curves of the vector field X generates a smooth bundle, which is called Hopf bundle .In the future, will require the following statement [2] Proposition. Vector fieldin R n is a Killing field if and only if the conditions are satisfied ∂ xi i ∂x j + ∂ξ j ∂x i = 0 i = j, ∂ξ i ∂x i = 0, i = 1, ..., n.
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