## The functor of Amari and Riemannian dynamics

07/11/2017
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## Résumé

Let E be a smooth vector bundle over a manifold M. Let ℝ the trivial bundle M × ℝ ⟶ M. The gauge group of E is denoted by G (E) . The Lie algebra of G (E) is denoted by ℊ(E). A gauge structure in E is a pair (E, ∇), where ∇ is a connection in E. A metric structure in E is a vector bundle homomorphism g∶ E x E ⟶ ℝ. A connection ∇ is called a metric connection in (E,g) if ∇g = 0.
Our propose is to discuss the question whether a giving connection is a metric connection. We use two approaches for answering this question. The first approach is based on the functor of Amari in E. This approach yells an numerical invariant Sb(∇). Another approach involves the group of isomorphism of the set of gauge structure generated by the set of metrics in E. We us this second approach for introducing a new numerical invariant index(∇). We show that both  Sb (∇) and index(∇) are characteristic obstruction to ∇ being a metric connection.
Loosely speaking, the following claims are equivalent: (1) The holonomy group of ∇ is an orthogonal subgroup, (2) Sb(∇) = 0, (3) index(∇)=0.

## Collection

The functor of Amari and Riemannian dynamics Michel Nguiffo Boyom, Ahmed Zeglaoui
Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

Let E be a smooth vector bundle over a manifold M. Let ℝ the trivial bundle M × ℝ ⟶ M. The gauge group of E is denoted by G (E) . The Lie algebra of G (E) is denoted by ℊ(E). A gauge structure in E is a pair (E, ∇), where ∇ is a connection in E. A metric structure in E is a vector bundle homomorphism g∶ E x E ⟶ ℝ. A connection ∇ is called a metric connection in (E,g) if ∇g = 0.
Our propose is to discuss the question whether a giving connection is a metric connection. We use two approaches for answering this question. The first approach is based on the functor of Amari in E. This approach yells an numerical invariant Sb(∇). Another approach involves the group of isomorphism of the set of gauge structure generated by the set of metrics in E. We us this second approach for introducing a new numerical invariant index(∇). We show that both  Sb (∇) and index(∇) are characteristic obstruction to ∇ being a metric connection.
Loosely speaking, the following claims are equivalent: (1) The holonomy group of ∇ is an orthogonal subgroup, (2) Sb(∇) = 0, (3) index(∇)=0.

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Let E be a smooth vector bundle over a manifold M. Let ℝ the trivial bundle M × ℝ ⟶ M. The gauge group of E is denoted by G (E) . The Lie algebra of G (E) is denoted by ℊ(E). A gauge structure in E is a pair (E, ∇), where ∇ is a connection in E. A metric structure in E is a vector bundle homomorphism g∶ E x E ⟶ ℝ. A connection ∇ is called a metric connection in (E,g) if ∇g = 0.

Our propose is to discuss the question whether a giving connection is a metric connection. We use two approaches for answering this question. The first approach is based on the functor of Amari in E. This approach yells an numerical invariant Sb(∇). Another approach involves the group of isomorphism of the set of gauge structure generated by the set of metrics in E. We us this second approach for introducing a new numerical invariant index(∇). We show that both  Sb (∇) and index(∇) are characteristic obstruction to ∇ being a metric connection.

Loosely speaking, the following claims are equivalent: (1) The holonomy group of ∇ is an orthogonal subgroup, (2) Sb(∇) = 0, (3) index(∇)=0.

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adfa, p. 1, 2011. © Springer-Verlag Berlin Heidelberg 2011 The functor of Amari and Riemannian dynamics NGUIFFO BOYOM Michel Alexander Grothendieck institute UMontpellier – France ZEGLAOUI Ahmed Alexander Grothendieck institute UMontpellier and USTHB –Algeria Abstract. Let E be a smooth vector bundle over a manifold M. Let ℝ ෩ the trivial bundle  ‫ܯ‬ × ℝ ⟶ ‫ܯ‬ . The gauge group of E is denoted by ࣡(‫)ܧ‬. The Lie algebra of ࣡(‫)ܧ‬ is denoted by ℊ(‫)ܧ‬. A gauge structure in E is a pair(‫ܧ‬, ∇), where ∇ is a connection in E. A metric structure in E is a vector bundle homomorphism ݃ ∶ ‫ܧ‬ × ‫ܧ‬ ⟶ ℝ ෩. A connection ∇ is called a metric connection in (‫ܧ‬, ݃) if ∇݃ = 0. Our propose is to discuss the question whether a giving connection is a metric connection. We use two approaches for answering this question. The first ap- proach is based on the functor of Amari in E. This approach yells an numerical invariant ‫ݏ‬௕(∇). Another approach involves the group of isomorphism of the set of gauge structure generated by the set of metrics in E. We us this second ap- proach for introducing a new numerical invariant index(∇). We show that both ‫ݏ‬௕(∇) and index(∇) are characteristic obstruction to ∇ being a metric connection. Loosely speaking, the following claims are equivalent: (1) The holonomy group of ∇ is an orthogonal subgroup, (2) ‫ݏ‬௕(∇) = 0, (3) index(∇)=0.