Subject: Answer to the question about diffeomorphisms
From: Vitali Kapovitch <vkapovit@gmail.com>
Date: Thu, 15 Jun 2006 18:36:56 -0400

> Let N be a smooth manifold, g and h two Riemannian structures on  it, 
and f :
> N ----> N a smooth.
> If f is homotopic to the identity and harmonic with respect to g on  the
> domain and h on the codomain is it a diffeomorphism?
>
> In dimension two and for N compact with non positive sectional  
curvature it
> is true, but for higher dimension?
>

This is false in high dimensions even for negatively curved  manifolds. 
Not sure about the earliest reference but see for example
MR2104792 (2005k:58026)
Farrell, F. T.(1-SUNY2); Ontaneda, P.(BR-FPN)
Harmonic cellular maps which are not diffeomorphisms.
Invent. Math. 158 (2004), no. 3, 497--513.

Vitali Kapovitch
___________________________________________________________________

Subject: Re: question about diffeomorphisms
From: "Prof. A. R. Shastri" <ars@math.iitb.ac.in>
Date: Fri, 16 Jun 2006 10:29:48 +0530 (IST)

Reply to the question by Boccellari:
Of course, one have to include the condition that the domain is compact, 
for the obvious reason that any constant map from n dimensional Eulcidean 
space to itself is harmonic as well as homotopic to identity. 

Anant R. Shastri