Geometry On The Unit Tangent Bundle T₁S～(2n+1)

In this paper, we give an explicit representation of the Sasaki metric on a vector bundle. In particular, we get Sasaki metric on unit tangent bundle T₁S²ⁿ⁺¹, by which we calculate the volume of the Hopf vector field V_h. From Gysin sequence, we get the cohomology of T₁S²ⁿ⁺¹. By Euler characteristic, we define a calibration on T₁S²ⁿ⁺¹ and show that the submanifold L²ⁿ⁺¹ is an integral submanifold of the calibration. We also show that Hopf vector field V_h is the integral submanifold only when n is equal to one. With a Sasaki metric defined by Hopf vector field, we show that the Hopf vector field has minimum volume on S²ⁿ⁺¹ for all n.