On a Riemannian submanifold whose slice representation has no nonzero fixed points

Taketomi, Yuichiro

doi:10.32917/hmj/1520478020

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…This together with (2.3) implies that the following diagram commutes: In general it is not clear that conversely the arid property of Φ −1 K (N) implies the arid property of N or not. However the next theorem shows that under suitable assumptions the arid property of Φ −1 K (N) is equivalent to that of N. To explain this we now introduce some terminologies which are used in a context slightly wider than [29]. Let M be a submanifold immersed in a finite dimensional Riemannian manifoldM .…”

Section: The Arid Propertymentioning

confidence: 99%

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Austere and arid properties for PF submanifolds in Hilbert spaces

Morimoto

2020

Differential Geometry and its Applications

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Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.2010 Mathematics Subject Classification. 53C40.

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Section: The Arid Propertymentioning

confidence: 99%

“…In [29] he gave an example of an arid submanifold which is not an austere submanifold (therefore not a weakly reflective submanifold). Also he showed that any isolated orbit of a proper isometric action is an arid submanifold.…”

Section: Introductionmentioning

confidence: 99%

Austere and arid properties for PF submanifolds in Hilbert spaces

Morimoto

2020

Differential Geometry and its Applications

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“…Remark 3. Recently Taketomi [9] introduced a generalized concept of weakly reflective submanifolds, namely arid submanifolds. The theorems and corollaries in this section are still valid in the arid case (see also [6]).…”

Section: Resultsmentioning

confidence: 99%

On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

Morimoto¹

2020

Preprint

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We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.2010 Mathematics Subject Classification. 53C40.

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“…Remark 8.4. Taketomi [26] introduced a generalized concept of weakly reflective submanifolds, namely arid submanifolds. Similarly to Theorem 8.1 we can prove that if N is an arid submanifold of G then Φ −1 (N) is an arid PF submanifold of V g .…”

Section: The Weakly Reflective Propertymentioning

confidence: 99%

On the geometry of orbits of path group actions induced by sigma-actions

Morimoto¹

2022

Preprint

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In the previous paper the author gave a formula for the principal curvatures of proper Fredholm (PF) submanifolds obtained through the parallel transport map over a compact symmetric space and showed an explicit formula for the principal curvatures of orbits of path group actions induced by Hermann actions. In this paper, introducing the concept of canonical isomorphism of path space, we derive a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map over a compact Lie group and show an explicit formula for the principal curvatures of orbits of path group actions induced by sigma-actions. It turns out that all known computational results of principal curvatures of PF submanifolds are special cases of the formula given in the previous paper. Moreover we study austere and weakly reflective properties of orbits of those path group actions and give new examples of infinite dimensional austere PF submanifolds and weakly reflective PF submanifolds in Hilbert spaces.

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On a Riemannian submanifold whose slice representation has no nonzero fixed points

Cited by 7 publications

References 10 publications

Austere and arid properties for PF submanifolds in Hilbert spaces

Austere and arid properties for PF submanifolds in Hilbert spaces

On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

On the geometry of orbits of path group actions induced by sigma-actions

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