Page 236 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 19: Serret-Frenet Formulae

10. Vertical projection from E onto its xy-plane is given by the map Notes
8
 : E  E : (x, y, z)  (x, y, 0).
8
8
A unit speed curve b : [0, L]  E lies above the xy-plane and has vertical projection
8
s s
8
  :[0,L]  E : s  ( cos(logs/2), sin(logs/2),0).

2 2
Find explicitly a suitable  and for it compute the Frenet-Serret frame, curvature and
torsion.

Answers: Self Assessment

1. moving frame along f 2. Frenet n-frame or Frenet frame
3. linear isometry 4. non-degeneracy conditions
5. X’(t) = –X(t)k(t)

19.6 Further Readings

Books Ahelfors, D.V. : Complex Analysis
Conway, J.B. : Function of one complex variable
Pati, T. : Functions of complex variable
Shanti Narain : Theory of function of a complex Variable
Tichmarsh, E.C. : The theory of functions

H.S. Kasana : Complex Variables theory and applications
P.K. Banerji : Complex Analysis
Serge Lang : Complex Analysis

H. Lass : Vector & Tensor Analysis
Shanti Narayan : Tensor Analysis
C.E. Weatherburn : Differential Geometry
T.J. Wilemore : Introduction to Differential Geometry
Bansi Lal : Differential Geometry.

LOVELY PROFESSIONAL UNIVERSITY 229