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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Basic Constructions and Examples
Erschienen in: Metric Foliations and Curvature
Verlag: Birkhäuser Basel
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Any Riemannian submersion can be used to generate new ones by deforming the metric in the vertical direction. To be specific, let π : (M, 〈, 〉 →
B
be a Riemannian submersion. Given
φ
:
M
→ ℝ, define a new metric 〈, 〉
φ
on
M
by
$$\left\langle {e,f} \right\rangle _\phi = e^{2\phi (p)} \left\langle {e^v ,f^v } \right\rangle + \left\langle {e^h ,f^h } \right\rangle , e,f \in M_p , p \in M. $$
Since the horizontal metric is unchanged, π : (
M
, 〈, 〉
φ
) →
B
is still a Riemannian submersion.
X, Y, Z
will denote basic fields, T
i
vertical ones, and
$$ \tilde \nabla $$
,
$$ \tilde R $$
the Levi-Civita connection and curvature tensor, respectively, of 〈, 〉
φ
. We will assume that the deformation is constant along fibers, or equivalently, that the gradient of φ is basic.