In conformal geometry, the conformal Killing equation on a manifold of space-dimension n with metric describes those vector fields which preserve up to scale, i.e.
for some function (where is the Lie derivative). Vector fields that satisfy the conformal Killing equation are exactly those vector fields whose flow preserves the conformal structure of the manifold. The name Killing refers to Wilhelm Killing, who first investigated the Killing equation for vector fields that preserve a Riemannian metric.
By taking the trace we find that necessarily . Therefore we can write the conformal Killing equation as
In abstract indices
where the round brackets denote symmetrization.
Famous quotes containing the words killing and/or equation:
“It is not worth the bother of killing yourself, since you always kill yourself too late.”
—E.M. Cioran (1911–1995)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)