Elementary Topics in Differential Geometry   Google Book

                                                                                                             John A. Thorpe

Solution to exercises up to 5 August, 2006 (Chapter 1 to 18, 22) in PDF.   

DISCLAIMER

           View in whole: Chapter 1-13: [pdf] (6.2 MB).       From Chapter 14 to 18, 22: [pdf] (5.2 MB).

           View by Chapter (showing the page # that each chapter's solution spans into on my notebook):

                               1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  18,  19,  20,  21,  22

           View by Page of solution on the my notebook:

                             1     2     3     4     5     6     7     8     9     10    11    12    13    14    15

                             16   17   18   19   20   21   22   23   24   25    26    27    28    29    30

                             31   32   33   34   35   36   37   38   39   40    41    42    43    44    45  

                             46   47   48   49   50   51   52

Schedule:     Planned finish date Real finished date Chapter # Pages Text Exercises Text Exercises 9 9 29 May 30 May  1 June1 2 June 10 6 31 May 31 May 3 June 3 June 11 14 13 June 14 June  14 June2 20 June3 12 13 15 June 16 June  24 June 26 June 13 13 18 June 19 June 1 July   4 July4 14 13 20 June 21 June 6 July  8 July 15 11 22 June 23 June  12 July  16 July5 16 7 24 June 24 June  17 July 18 July6 17 17 26 June 27 June 23 July 25 July 18 7 28 June 28 June  27 July 28 July 19 14 30 June 1 July     20 13 2 July 3 July     21 20 6 July 7 July     22 10 8 July 9 July 3 Aug 5 Aug 23 11 10 July 11 July     24 14 13 July 14 July

1. Delay due to NICTA TechFest from 30 May to 31 May, 2006.

2. Delay due to NIPS 2006 submission deadline (10 June, 2006 Sydney time).

3. My God.  One week for one chapter's exercise.  Then the whole book can take me a century (if I could live that long)!

4. The Exercise 13.4 seems wrong.  It cost me 4 days to prove, but finally I found a counter-example.  Please contact me if my answer is wrong.  Thank you.

5. The Exercise 15.4 cost me 3 days :(  I hate looking for counter-examples! I would rather prove something...

6. Found two flaws in the exercises of the book.  Both questions Ex 16.3 and Ex 16.4 forgot to assume that curvature is positive.

Resources online:

A collection of definitions in this textbook, written by David Conner.  Covering Chapter 1 to Chapter 15, and 17.  I will write the rest.

Errata: (only important entries are mentioned)

Page 48:

Original: Fourth line above the Corollary: "Hence the vector field on $I \cup \tilde I$ which is equal to V on I and to W on $\tilde I$ extends V to a solution of ...".

It should be: Hence the vector field on $\tilde I \cup J$ which is equal to V on $\tilde I$ and to W on J extends V to a solution of ...".

Page 55:

Original: Second line in Ex. 9.15 (a) "Hint: show that The equation near the middle of the page: \nabla _v N = - (N\dot \circ \alpha )(t_0)
It should be: - \nabla _v N = - (N\dot \circ \alpha )(t_0)

Page 61:

Original: The equation near the middle of the page: \tilde \alpha (t) = X(\tilde \alpha (t))
It should be: \dot \tilde \alpha (t) = X(\tilde \alpha (t))

Page 66:

Original: Last equation in Ex. 10.6: (N \circ \alpha ) = - (\kappa \circ \alpha )T

It should be: (N\dot \circ \alpha ) = - (\kappa \circ \alpha )T

Page 71:

Original: 8th line: \beta (t) = \alpha (t - t_1) = \alpha ( (t \tau) - t_2) = \beta (t + \tau)

It should be: \beta (t) = \alpha (t - t_1) = \alpha ( (t + \tau) - t_2) = \beta (t + \tau)

Page 84:

Original: The second line from the bottom in the proof of Theorem 1: k(v) = (i\dot \circ \alpha )(t_0 ) \cdot N(p) = ...

It should be: k(v) = (i\ddot \circ \alpha )(t_0 ) \cdot N(p) = ...

Page 96:

Original: Last line of caption of Figure 13.1: H(p) = \{ q \in R^{n + 1} |q \cdot N(p) = 0\}

It should be: H(p) = \{ q \in R^{n + 1} |(q - p) \cdot N(p) = 0\}

Page 162: (exists only in old prints of the book)

Original: Exercise 18.7:    ... Verify 

They should be:   ... Verify 

Page 212:

Original: Line 12: then {\psi(e_1), ..., \psi(e_{n+1}} is also an orthonormal ...

They should be:  then {\psi_1(e_1), ..., \psi_1(e_{n+1}} is also an orthonormal ...

Page 216:

Original: Line 13: {X_1(t), ..., X_{n+1}(t)} is an orthonormal basis ...

They should be:  {X_1(t), ..., X_n(t)} is an orthonormal basis ...