A vector field X on a Riemannian manifold $(M,g)$ is 2-Killing if $L_XL_Xg=0$. This vector field is a generalization of Killing vector fields. There are two articles in the literature review considering the geometric interpretation of this concept.
1. http://www.emis.de/journals/BJGA/v13n1/B13-1-OP.pdf
2. http://www.worldscientific.com/doi/abs/10.1142/S0129167X15500652
Now, I want to enlarge the class of conformal vector fields in the same line.
A vector field X is 2-conformal on a Riemannian manifold $(M,g)$ if
$L_XL_Xg=\rho g$
My question is: What is the geometric interpretation of this vector field? Also, Does the left hand side mean double dragging of the metric, or some thing else?
Dear Sameh, since $L_Xg$ is a 2 covariant tensor, I recommend to you to study the properties of $L_XT$ in the book of John Lee "Introduction to Smoth Manifolds" (see chapter of Lie Derivatives). You can computation some direct formulas for this particular case and to test in the manifolds which are Einstein, especially when you teste such formulas in the Euclidean sphere you can obtain some geometrical conclusions. Moreover, if you want speak with me another time, I suggest to you send an email to [email protected]
I am sorry for my english
Best regards
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