(b) ([10%])be a family of plane curves defined by
 where.Show that Ir intersects at exactly one point for each  and the angle between the tangent vectors of  andΓ_T at the intersection point is independent of T.

(a) ([10%]) Find the curvature of where x is the curvature and n is the unit normal vector on y and

(2) ([25%]) Prove that if a regular surface in R3 contains a straight line, then the surface has non-positive Gauss curvature at all the points of this line.

(3) ([30%]) Consider a regular surface S where and the inversion Φ :R3＼  given by Compare the mean curvature and the Gauss urvatures of S and of the image surface S' = Φ(S). More precisely, show that and where N(p) is the unit normal of S at p, H'(Φ(p) is the mean curvature of s' at Φ(p), K'(Φ(p) is the Gauss curvature of S' at Φ(p).

(4) ([25%]) Prove that if S is a regular surface that is diffeomorphic to a cylinder and has Gaussian curvature K< O, then S has at most one simple closed geodesic (up to reparametrization).

110年 - 110 國立臺灣大學_碩士班招生考試_數學研究所:幾何#101955