By Maria Falcitelli

ISBN-10: 9812388966

ISBN-13: 9789812388964

ISBN-10: 9812562338

ISBN-13: 9789812562333

For graduate scholars and researchers attracted to differential geometry and theoretical physics, Falcitelli, Anna Maria Pastore (both U. of Bari, Italy) and Steve Ianus (U. of Bucharest, Romania) supply a scientific exposition of the elemental instruments at the concept of Riemannian submersions and its advancements. Submersions are specified instruments in differential geometry, but additionally play a job in Riemannian geometry, particularly while the manifolds concerned hold an extra constitution. they're in a feeling a counterpart of immersions, however the corresponding theories and histories are really assorted.

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**Example text**

Let 7r : ( M , g ) + ( B , g ' ) be a Riemannian submersion, with M connected and complete. Then B is complete. Moreover, if the fibres are totally geodesic, then IT acts as the projection of a bundle associated with a principal fibre bundle with structure group the Lie group of isometries of the fibre. Proof. To prove the geodesic-completeness of B , one considers a geodesic c : [a,b] -+ B. Putting c(a) = x, for any p E 7r-'(x) the horizontal lift y of c starting at p is a geodesic on M . Indeed, the basic vector field V,?

1 provide examples of the above projections. Moreover, we explicitly describe an S7-fibration from S15 on the sphere S8($). The representation theory of Clifford algebras is applied in Sec. e. to prove that, up to equivalence, the previous examples are the only Riemannian submersions with totally geodesic fibres and total space S”. In the last Sec. 5, we discuss the minimality and the totally geodesicity of the submanifold obtained lifting a submanifold of the base space to the total space. Then, via the Hopf submersion, we construct a Riemannian submersion p : P2,+1(C) + Pn(Q) with fibres isometric to P1(C).

Eschenburg constructs an interesting class of compact manifolds which are realized as biquotients, i. e. as the base space of a homogeneous principal bundle. More precisely, let G be a compact Lie group and K a closed subgroup of G. The map ((a,k ) , a') EGxK xG aa'k-l E G , defines a transitive action of Go = G x K on G and the isotropy group of the unit element is A K = { ( k , k ) I k E K } . Consider a bi-invariant metric < , >o on G and the induced metric on K . This gives a bi-invariant metric on G x K , denoted again by <, >o.

### Riemannian Submersions and Related Topics by Maria Falcitelli

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