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Let   Z   be a quadratic hypersurface ofP  n    (  R  )defined over   Q   containing points whose coordinates are linearly independent over   Q  . We show that, among these points, the largest exponent of uniform rational approximation is the inverse    1  /  ρ    of an explicit Pisot number    ρ  <  2    depending only on   n   if the Witt index (over   Q  ) of the quadratic form   q   defining   Z   is at most 1, and that it is equal to 1 otherwise. Furthermore, there are points of   Z   which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound    1  /  ρ    uses a recent transference inequality of Marnat and Moshchevitin. In the case    n  =  2   , we recover results of the second author while for    n  >  2   , this completes recent work of Kleinbock and Moshchevitin.

Rational approximation to real points on quadratic hypersurfaces