Page 354 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 28: Local Intrinsic Geometry of Surfaces

3. Let : U   U be a diffeomorphism, and let Y be a vector at u U.  Then for any smooth Notes


function f on U, we have: .................
4. The ................. cannot be used as an intrinsic directional derivative of a vector field Z,
which should only depend on the direction vector Y at a single point.
5. Let the Riemannian metric g be induced by the parametric surface X. Then the image

under dX of the covariant derivative ................. is the projection of  i Z onto the tangent
space.
6. The Laplacian of f if the divergence of the gradient of f: .................

28.7 Review Questions

1. Prove that a parametric surface X : U   is conformal if and only if its first fundamental
3
form g is conformal to the Euclidean metric  on U.
2. Two Riemannian metrics g and g  on an open set U   are conformal if g  e g for
2
2

some smooth function .
Answers: Self Assessment

1. Riemannian metric 2. Euclidean metric.

3.   d Y        Y  f  . 4. Lie derivative
f 

5. dX  i  Z 6. f = div f.

28.8 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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