Differential Geometry I, WS 2020/21

Correction log for the lecture notes:

Date	Page	Correction
11.11.2020		added missing reference to Lee's book on smooth manifolds
11.11.2020	p. 22	Definition 1.2.1: last part of the defintition: changed x to y.
12.11.2020	p. 22	diagram: corrected label of left vertical arrow
19.11.2020	p. 40	Corollary 1.3.7: added missing condition (dim M is non-zero)
19.11.2020	p. 42/43	Example 1.3.15: corrected the definition of the warped product
26.11.2020	p. 41	Example 1.3.9: longer -> shorter
26.11.2020	p. 48	Proposition 1.4.7: Added connectedness assumption.
26.11.2020	p. 49	line 2: introduced x^1, ..., x^n
26.11.2020	p. 49	Proposition 1.4.10: \S^0 is not connected (thus not symmetric)
26.11.2020	p. 50	proof of Proposition 1.4.10: corrected definition of N
03.12.2020	p. 64	Example 1.4.25: radius corrected
03.12.2020	p. 63	Example 1.5.4: typos corrected
03.12.2020	p. 67	Corollary 1.5.12: added the claim that the closure of V is contained in U (this is satisfied by construction and improves applicability of the statement).
03.12.2020	Week 6	Exercise 5.2: corrected the definition of the action for the Klein bottle
04.12.2020	p. 74	Proposition 1.5.27: added missing det
09.12.2020	p. 71	Remark 1.5.14: corrected the integration formulae (missing inverse of the chart)
10.12.2020	p. 85/86	Remark 2.1.12: corrected two typos in the formulae (missing ^U and missing sum)
11.12.2020	p. 94	Definition 2.2.31: removed confusing condition "s < t".
17.12.2020	p. 93	Remark 2.2.28: corrected indices in the parallel transport equation
21.12.2020	Week 8.5	curves: cleaned up the first sentence
23.12.2020	p. 87	Figure 2.2: Adobe Reader couldn't handle this; this should be fixed now.
07.01.2021	p. 83	Example 2.2.8: Vector fields cannot necessarily be extended to all of \R^N if M is not closed. The statements are corrected accordingly.
07.01.2021	p. 106	ungarbled the computation in the proof of part 3.
08.01.2021	Week 9	Exercise 8.2: Added missing parentheses in the definition of T
13.01.2021	p. 111	Definition 2.4.4: corrected target type of Rm
13.01.2021	p. 111	Remark 2.4.5: small clarification
13.01.2021	p. 112f	Proposition 2.4.7: M_1 instead of M; added missing composition with phi.
13.01.2021	p. 114	proof of 2.: typo
18.01.2021	p. 138--141	corrected several of the differentiation indices
20.01.2021		corrected several typos; mainly concerning Chapter 2.5 and 3.1.1
21.01.2021	p. 136	Proposition 3.1.14: added condition to uniformly normal neighbourhoods (as in the lecture)
21.01.2021	p. 140	proof of Proposition 3.2.7: the proof starts now with choosing a Riemannian metric on M
24.01.2021	p. 124	Theorem 2.5.5; formula in part (1): the last part is corrected.
31.01.2021	Week 12	Exercise 11.3: Corrected/clarified hypothesis on the curve gamma.
31.01.2021	p. 159	Remark 3.5.1: Corrected/clarified hypothesis on the curve gamma.
03.02.2021	Week 12	Exercise 11.4: "symmetric" instead of "locally symmetric"; as this is on very short notice: If you already worked on this problem and showed that the previous formulation was incorrect, then this is also fine!
04.02.2021	p. 169	proof of Theorem 3.5.15: added missing signs
04.02.2021	p. 164	proof of Proposition 3.5.7: corrected the blunder (as discussed in the lecture)
10.02.2021	p. 178	typos in the proof of Theorem 4.3.2 corrected