1.7. Invariance Properties of Flows 35 the inner product) and hence if and only if the matrix ∂Vi ∂xj is every- where skew-symmetric. Show that isometric flows are also measure preserving. Exercise 1–27. Show that the translation flows generated by con- stant vector fields are isometric and also that the flow generated by a linear vector field V (x) = Ax is isometric if and only if A is skew- adjoint. Conversely show that if V (x) is a vector field generating a one-parameter group of isometries of Rn, then V (x) = v + Ax, where v is a point of Rn and A is a skew-adjoint linear map of Rn. Hint: Show that ∂2Vi ∂xj∂xk vanishes identically.
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