9

I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space $M = G/H$ the Euler characteristics is non-negative.
Mostow, G.D.: The extensibility of local Lie groups of transformations and groups on surfaces. Ann.
Math. (...

6

Oh, I happen to know the guy who wrote that PDF. As to your questions.
Yes, that's the idea. In $V \times A$, $F = f_0$ so the critical points are in one-to-one correspondence with those of $f_0$. The shift in degree comes from this $g$ which has a maximum at $0$. Similarly in $V \times B$ but there is no shift in degree.
Intuitively, the way $g$ and $\...

4

A simple example where the answer is 'no' is when $M=\mathbb{RP}^2$ (with, say, the standard metric of Gauss curvature $K\equiv1$, though, in dimension $2$, only the conformal structure on $M$ matters in the definition of harmonic map).
There is no non-constant harmonic map $f:\mathbb{RP}^2\to S^2$ (when $S^2$ given the standard metric with $K\equiv1$). In ...

4

If a domain $\Omega$ has boundary of class $C^k$, $k\geq 2$, then in fact the distance function $d$ to the boundary of $\Omega$ is of class $C^k$ in a neighborhood of the boundary. This is exactly what is proved in Lemma 14.16 in [1] mentioned by OP.
In the case of a convex set we have the following result. Note that we have a better regularity than the one ...

fa.functional-analysis dg.differential-geometry mg.metric-geometry differential-topology set-valued-analysis

3

Response to the first question:
Pantilie, Radu, A simple proof of the de Rham decomposition theorem, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 36, No. 3-4, 341-343 (1992). ZBL0811.53040.

dg.differential-geometry mg.metric-geometry sheaf-theory at.algebraic-topology gt.geometric-topology

2

Such a manifold has bounded diameter, and is therefore compact. To see this, let $0 < T < \infty$ be the quantity you define. I claim that $\mathrm{diam} M \leq T$.
Additionally for each point $p \in M$ and unit tangent vector $v \in T_p M$ let $\tau(v)$ be the first conjugate time along the geodesic $\gamma: t \mapsto \exp_p(tv)$. Explicitly, $\tau(v) ...

2

I'll use $\Omega \in \Omega^2(P,\mathfrak{g})$ to denote the curvature tensor of $\omega$.
One way of identifying these two expressions is through Cartan's structure equation
$$\Omega = d\omega + \frac{1}{2}\omega \wedge \omega.$$
A reference is Kobayashi-Nomitsu's book, Chapter II.5;
here we use the convention
$$d \omega(X,Y) = \frac{1}{2}(X \cdot \omega(Y) ...

1

"The averaging process" which is used in [Chevalley-Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124] on pages 90-92, may be used not only for compact Lie groups, but almost literally in the same form for any homogeneous spaces of such Lie groups.
In more general situation (for arbitrary linear ...

1

The following theorem is cloasely related to your second question.
It is proved in The de Rham decomposition theorem for metric spaces by T. Foertsch and A. Lytchak.

dg.differential-geometry mg.metric-geometry sheaf-theory at.algebraic-topology gt.geometric-topology

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