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Consider a Riemannian manifold \( (M,g)\). Let \(\pi\colon TM\to M\)
be a natural projection. The Levi-Civita connection \(\nabla\) of
\(g\), gives a natural splitting \(T(TM)= H\oplus V \) of the second
tangent bundle \(\pi_*\colon T(TM) \to TM\), where the vertical
distribution \(V\) is the kernel of \(\pi_*\), and the horizontal
distribution \(H\) is the kernel of, so called, connection map
\(K\). We say that a Riemannian metric \(h\) on \(TM\) is
natural if \(\pi \colon (TM,h)\to (M,g)\) is a Riemannian
submersion (with respect to the splitting \(T(TM)= H\oplus V \) ).
In the talk we introduce some special class of natural metrics,
called Cheeger-Gromoll type metrics. Next we give an answer to the
following problem:
Let \(\varphi\colon (M,g) \to (M',g')\) be a smooth map between
Riemannian manifolds. Equip tangent bundles \(TM\) and \(TM'\) with
Cheeger-Gromoll type metrics \(h\) and \(h'\), respectively. When
\(\Phi=\varphi_*\colon (TM,h)\to (TM',h')\) is conformal?
Interesting enough, there is an essential difference between the
cases \(\dim M=2\) and \(\dim M \geq 3\). We show that in the second
case \(\Phi\) is conformal if and only if \(\varphi\) is a homothety
and totally geodesic immersion and some special relations between
\(h\) and \(h'\) hold. In this case \(\Phi\) is also a homothety
with the same dilatation as \(\varphi\). However, in the first case
it may happen that \(\Phi\) is conformal, although \(\varphi\) is
not a totally geodesic immersion. Then \(\Phi\) is no longer a
homothety. An example of such a map is given.
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