scholarly journals A Decomposition Theorem for Immersions of Product Manifolds

2015 ◽  
Vol 59 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Ruy Tojeiro
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Type Theorem ◽  
Decomposition Theorem ◽  
Product Manifold ◽  
Isometric Immersions ◽  
Product Metrics ◽  
Special Cases ◽  
De Rham ◽  
Shape Operators

AbstractWe introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally or globally decomposed as a product manifold endowed with a polar metric. For such a product manifold, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapted to its product structure in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as of its extension by Nölker to isometric immersions of warped products.

scholarly journals A de Rham decomposition type theorem for contact sub-Riemannian manifolds

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Marek Grochowski
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Parallel Transport ◽  
Holonomy Group ◽  
Decomposition Theorem ◽  
Reeb Vector Field ◽  
Indefinite Metrics ◽  
De Rham ◽  
Riemannian Case

AbstractIn this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$ ξ is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$ q ∈ M such that the holonomy group $$\Psi (q)$$ Ψ ( q ) acts reducibly on H(q) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$ H ( q ) = H 1 ( q ) ⊕ ⋯ ⊕ H m ( q ) into $$\Psi (q)$$ Ψ ( q ) -irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$ H = H 1 ⊕ ⋯ ⊕ H m of H into sub-distributions $$H_i$$ H i . Unlike the Riemannian case, the distributions $$H_i$$ H i are not integrable, however they induce integrable distributions $$\Delta _i$$ Δ i on $$M/\xi $$ M / ξ , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$ T ( U / ξ ) = Δ 1 ⊕ ⋯ ⊕ Δ m , and the latter decomposition of $$T(U/\xi )$$ T ( U / ξ ) induces the decomposition of $$U/\xi $$ U / ξ into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

SEMI-INVARIANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

2011 ◽  
Vol 08 (07) ◽  
pp. 1439-1454 ◽  
Author(s):  
BAYRAM ṢAHIN
Keyword(s):  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Decomposition Theorem ◽  
Necessary And Sufficient Conditions ◽  
Classification Theorem ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Riemannian Map ◽  
Necessary And Sufficient

This paper has two aims. First, we show that the usual notion of umbilical maps between Riemannian manifolds does not work for Riemannian maps. Then we introduce a new notion of umbilical Riemannian maps between Riemannian manifolds and give a method on how to construct examples of umbilical Riemannian maps. In the second part, as a generalization of CR-submanifolds, holomorphic submersions, anti-invariant submersions, invariant Riemannian maps and anti-invariant Riemannian maps, we introduce semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds, give examples and investigate the geometry of distributions which are arisen from definition. We also obtain a decomposition theorem and give necessary and sufficient conditions for a semi-invariant Riemannian map to be totally geodesic. Then we study the geometry of umbilical semi-invariant Riemannian maps and obtain a classification theorem for such Riemannian maps.

Symmetric-like Riemannian manifolds and geodesic symmetries

1995 ◽  
Vol 125 (2) ◽  
pp. 265-282 ◽  
Author(s):  
Jürgen Berndt ◽  
Friedbert Prüfer ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Jacobi Operators ◽  
Geodesic Spheres ◽  
Shape Operators

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

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