Page 395 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 30.11 Self Assessment

1. The computation is a bit involved, and it will lead us to the ............... , introduced in 1869.
2. The quantity K = k k called the ............... and the quantity H = (k + k )/2 called the mean
1
1 2
2
curvature, play a very important role in the theory of surfaces.
3. At a parabolic point, one of the two principal curvatures is ..............., but not both. This is
equivalent to K = 0 and H ¹ 0. Points on a cylinder are parabolic.
4. At a planar point, k = k = 0. This is equivalent to K = H = 0. Points on a plane are all planar
1
2
points! On a monkey saddle, there is a planar point. The principal directions at that point
are ...............
5. The derivative dN of the ............... at p measures the variation of the normal near p, i.e.,
p
how the surface “curves” near p.
6. The ............... of dN in the basis (X , X ) can be expressed simply in terms of the matrices
v
u
p
associated with the first and the second fundamental forms (which are quadratic forms).
30.12 Review Questions
1. Explain the Gauss Map and its Derivative dN.
2. Define the Dupin Indicatrix.
3. Describe the theorema Egregium of Gauss, the Equations of Codazzi-Mainardi, and Bonnet’s
Theorem.
4. Define Lines of Curvature, Geodesic Torsion, Asymptotic Lines.

Answers: Self Assessment

1. Christoffel symbols 2. Gaussian curvature
3. zero 4. undefined.
5. Gauss map 6. Jacobian matrix

30.13 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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