Page 346 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 27: Principal Curvatures

27.7 Self Assessment Notes

1. All curves with the same tangent vector will have the same ..................., which is the same
as the curvature of the curve obtained by intersecting the surface with the plane containing
T and u.
2. ................... is however in fact an intrinsic property of the surface, meaning it does not
depend on the particular embedding of the surface; intuitively, this means that ants living
on the surface could determine the Gaussian curvature.
3. Gaussian curvature only depends on the ................... of the surface.

4. An ................... definition of the Gaussian curvature at a point P is the following: imagine
an ant which is tied to P with a short thread of length r
5. A ................... corresponds to the inverse square radius of curvature; an example is a sphere
or hypersphere.

27.8 Review Question

1. Define Principal curvatures.
2. Discuss the concept of two dimensions.
3. Describe the Curvature of surfaces.
4. Explain the Mean curvature.
5. Explain the Higher dimensions: Curvature of space.

Answers: Self Assessment

1. normal curvature 2. Gaussian curvature

3. Riemannian metric 4. Intrinsic
5. positive curvature

27.9 Further Reading

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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