Page 227 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 13. The total twist plays an important role in modern molecular biology, especially with
respect to the structure of DNA.
14. Let x be the circle x(t) = (r cos(t), r sin(t), 0), where r is a constant > 1. Describe the collection
of points x(t) + z(t) where z(t) is a unit normal vector at x(t).

15. Prove that the total twist of a closed curve not passing through the origin is the same as the
total twist of its image by inversion through the sphere S of radius r centered at the origin.

16. Prove that the equations E (t) = q (t)E(t) can be written E (t) = d(t) × E (t), where d(t) =
'
'
i
i
j
i
ij
q (t)E (t) + q (t)E (t) + q (t)E (t). This vector is called the instantaneous axis of rotation.
23
12
3
2
1
31
17. Under a rotation about the x -axis, a point describes a circle x(t) = (a cos(t), a sin(t), b). Show
3
that its velocity vector satisfies x’(t) = d × x(t) where d = (0, 0, 1). (Compare with the
previous exercise.).
.
18. Prove that (v v)(w·w)”(v·w)2 = 0 if and only if the vectors v and w are linearly dependent.
19. Draw closed plane curves with rotation indices 0, –2, +3 respectively.
20. The theorem on turning tangents is also valid when the simple closed curve C has “corners.”
Give the theorem when C is a triangle consisting of three arcs. Observe that the theorem
contains as a special case the theorem on the sum of angles of a rectilinear triangle.
21. Give in detail the proof of the existence of  = (p ) used in the proof of the theorem on
0
turning tangents.  =  (p ) .
0
Answers: Self Assessment
1. straight line 2. tangential indicatrix
3. Fenchel’s Theorem 4. non-inflectional
5. w(t) = 0
18.14 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.

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