(i) Prove that the Veronese map V2: R 3 →R 6 defined such that Induces a homeomorphism of RP2 on toV2(S2).Show that V2(S2)is a subset of the hyper plane x 1 +x 2 +x 3 =1 in R6, and thus that RP2 is...

(i) Prove that the Veronese map V2: R³→R⁶
defined such that

Induces a homeomorphism of RP2 on toV2(S2).Show that V2(S2)is a subset of the hyper plane x¹+x²+x³=1 in R6, and thus that RP2 is home omor phics to a sub set of R⁵.Prove that this homeomorphism is smooth.

(ii)Prove that the Veronese map V3: R4→R10 defined such that

Induces

Ahomeomorphism of RP3 onto V3(S3). Show that V3(S3) is a subset of the hyper plane x1+x2+x3+x4=1 in R10, and thus that RP3 is homeomorphic to a subset of R9. Prove that this homeo morphism is smooth.