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Harmonic morphisms between Riemannian manifolds
Paul Baird and John C. Wood
London Mathematical Society Monographs, No. 29,
Book home page
Oxford University Press (2003)
Chapter 1, Proposition 1.2.1 is incorrect as stated, in particular,
equivalence between equations (1.1.1), (1.1.3) and equations (a) and
(b) is not guaranteed. The text after eqn (1.2.1) should read: If G satisfies
(a) and (b) at all points of A, then any smooth solution &phi : U
-> C satisfies (1.1.1) and (1.1.3), i.e., &phi is a harmonic morphism; furthermore, it is
submersive. In particular, this is all
that is required in the applications that follow. The proof is
flawed at equation (1.2.6), which does not hold in general.
Also, in equation (1.2.5), there should be a 2 multiplying the second term.
We remark that all submersive harmonic morphisms &phi : U -> C
are given in this way: simply take G(x,z) = z - &phi(x).
Chapter 1, proof of Lemma 1.3.3.
After the display (1.3.12), the roles of the indices 2 and 3 should be reversed.
Chapter 2, page 53, line 2, displayed equation. x0 should be x1.
Chapter 5, page 145, right-hand side of equation (5.2.5). As x is regarded as
a column vector, the transpose sign should be on the first x not the second.
Chapter 5, page 148, first line of Example 5.2.8. For `polynomial' read `non-polynomial'!
Chapter 5, page 158, 7th line of the proof of Theorem 5.5.7. For `Im' read `Im/2'.
Chapter 5, page 160, line 7 For `m = 2k&delta(n)' read `m = k&delta(n)'.
Chapter 12, page 389, Note 5 to Section 12.1, line -3. Insert `first' before `Pontryagin number'.
Chapter 13, page 421, middle, five lines before the end of Note 1. For `m &le 7' read `m &le 8'.
Chapter 14, Notes and comments for Section 14.3, note 3. Parmar (1991a,b) proved the result later found by Bejan and Benyounes (1999); in the latter paper, the Kähler condition is relaxed somewhat.
Chapter 14, Notes and comments for Section 14.6, note 2. That the Hopf map in Example 14.6.5 and the map in Example 14.6.6 are harmonic morphisms was established by Parmar (1991a,b).
Appendix A1, page 458, Definition A.1.3. In the display, the integral should be over Rm not K.
Appendix A2, page 462 line 8. For `Schwarz 1966' read `Schwartz 1998'.