41. Then choosing any smooth positive quadratic form on H gives a sub-Riemannian metric on the group. 42. In 1986, Margulis gave a complete resolution of the Oppenheim conjecture on quadratic forms and diophantine approximation. 43. Binary quadratic forms were considered already by Fermat, in particular, in the question of Eisenstein ). 44. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form . 45. For quadratic forms over a number field, there is a Hasse invariant ? for every finite place. 46. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. 47. Then the Arf invariant of this quadratic form can be used to distinguish the two extra special groups. 48. With additional structure, this ?-symmetric form can be refined to an ?-quadratic form . 49. For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant. 50. :: : The quadratic form can be determined from the fixed points of the order three elements.
