Families Of Such Surfaces Research Articles

Spherical chen surfaces which are mass-symmetric and of 2-type

The main purpose is to classify spherical Chen surfaces which are mass-symmetric and of 2-type. We prove that these surfaces are either pseudoumbilical or flat,moreover,in the later case,they lie fully in a 3-sphere,a 5-sphere,or a 7-sphere. Our result divides the family of such surfaces into four classes. In the last section, we give examples of surfaces which belong to each of these four classes.

Motion of magnetic lines of force

It is often said that magnetic lines of force in a conducting fluid move with the fluid. In the case of a plasma, this means that the lines of force move with the particle drift velocity 1 v p = ( E × H) H 2 . Such statements are not directly verifiable, since the velocity of a line of force is not a measurable quantity. However, the statement that the lines of force move with a certain velocity v(r, t) does have verifiable consequences, such as: (1) The flux through a closed curve moving with velocity v is constant. (2) A line, moving with velocity v, which is initially a line of force, remains a line of force in the course of its motion. Statement (2), as well as all other verifiable consequences of the original hypothesis, follows from statement (1). A velocity is said to be flux-preserving if it satisfies (1), line-preserving if it satisfies (2). It is permissible to ascribe a velocity v to the lines of force if and only if ∇ × (E + v × H) vanishes identically. It is always possible to choose a v satisfying this relation, although it is not generally possible to do this uniquely. To say that a certain velocity v is permissible means that all the verifiable consequences of ascribing this velocity to the lines of force are valid, i.e., that v is flux-preserving. In the case of the particle drift velocity the condition for flux-preservation reduces to ∇ × [ H(E·H) H 2 ] = 0 . Even if v p is not flux-preserving, there may be some closed curves moving with velocity v p which have constant flux. A semiexhaustive enumeration of such curves is given for a general electromagnetic field. Among these curves are those which lie in a surface everywhere perpendicular to H, if this surface is independent of time. A family of such surfaces will exist if and only if H·∇ × H and H × Ḣ both vanish identically. A velocity may be line-preserving without being flux-preserving, but not vice versa. The necessary and sufficient condition for line-preservation is that H × [∇ × (E + v × H)] should vanish identically. The motion of the lines of force in a plasma is related only to the transverse motion of the charged particles. The latter is separable from the longitudinal motion if and only if v p is line-preserving. A necessary and sufficient condition is also given for the separability of only one component of the transverse motion. The concept of a line of force is not relativistically covariant, because each point of a line of force has the same time coordinate. A curve in space-time which appears as a line of force in one frame of reference will therefore not be a line of force in another frame of reference. However, a moving line of force will trace out a two-dimensional surface in space-time, and it may be that this surface will intersect every space-like hyperplane in a line of force. In that case the surface will appear as the path of a moving line of force in every frame of reference, thus defining a moving line of force as a covariant concept. It is shown that a family of such surfaces exists if and only if E·H vanishes identically, in which case they will be generated by lines of force moving with the particle drift velocity v p .