In particular, the three spatial Killing vector fields have exactly the same form as the three nontranslational Killing vector fields in a spherically symmetric chart on E 3; that is, they exhibit the notion of arbitrary Euclidean rotation about the origin or spherical symmetry.
42.
In Minkowski space-time, in pseudo-Cartesian coordinates ( t, x, y, z ) with signature ( +,-,-,-), an example of Killing horizon is provided by the Lorentz boost ( a Killing vector of the space-time)
43.
For three dimensions, this gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one may use linear combinations of these to find three everywhere non-vanishing Killing vector fields \ xi ^ { ( a ) } _ { i },
44.
For three dimensions, this gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one may use linear combinations of these to find three everywhere non-vanishing Killing vector fields \ xi ^ { ( a ) } _ { i },
45.
When we said above that \ partial _ \ phi is a Killing vector field, we omitted the pedantic but important qualifier that we are thinking of \ phi as a " cyclic " coordinate, and indeed thinking of our three spacelike Killing vectors as acting on round spheres.
46.
When we said above that \ partial _ \ phi is a Killing vector field, we omitted the pedantic but important qualifier that we are thinking of \ phi as a " cyclic " coordinate, and indeed thinking of our three spacelike Killing vectors as acting on round spheres.
47.
It is obvious from the generators just given that the slices x = x _ 0 admit a SL ( 2, "'R "') action, and the slices t = t _ 0 admit a Bianchi III ( c . f . the fourth Killing vector field ).
48.
In a given spin manifold, that is in a Riemannian manifold ( M, g ) admitting a spin structure, the Lie derivative of a spinor field \ psi can be defined by first defining it with respect to infinitesimal isometries ( Killing vector fields ) via the Andr?Lichnerowicz's local expression given in 1963:
49.
With Y . Li, Liu introduced a new geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature; proved the global existence of the solution; discussed the global convergence of its solution and possible applications to the Hopf conjecture, as well as its relation to the Navier-Stokes equations on manifolds.
50.
Taking c = 0 in the defining property of a homothety, it is seen that every Killing is a homothety ( and hence an affine ) and the set of all Killing vector fields on M forms a Lie subalgebra of H ( M ) denoted by K ( M ) ( the "'Killing algebra "') and satisfies for connected " M"
