(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.

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

(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  andΓT at the intersection point is independent of T.

(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