Symmetric Spaces Rolling on Flat Spaces

Jurdjevic, Velimir; Markina, Irina; Leite, F. Silva

doi:10.1007/s12220-022-01179-5

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…Assume that G/H is a pseudo-Riemannian symmetric space. Then, by [1] (Sec. 4.2), a rolling of p over G/H along a given rolling curve can be viewed as a triple α(t), α(t), A(t) , where…”

Section: No-go Lemmamentioning

confidence: 97%

“…Remark 3. The discussion in [1] (Sec. 3) reveals that a rolling in the sense of Definition 6 is closely related to the classical definition of rolling in [8] (Ap.…”

Section: Definition 6 (Extrinsic Rolling (Ii))mentioning

confidence: 99%

“…Motivated by [1] (Prop. 3), we relate the two different notions of extrinsic rolling from Definitions 5 and 6.…”

Section: Definition 6 (Extrinsic Rolling (Ii))mentioning

confidence: 99%

“…whose solution is the horizontal lift of the development curve α(t) = π(q(t)) through q(0) = e. Note that in [1], G/H is always rolled over p, while in our work we consider p rolling over G/H. This choice is more convenient for us, because there is no need to invert q(t), as in [1] (Eq. 26).…”

Section: No-go Lemmamentioning

confidence: 99%

“…Although these definitions apply to general pseudo-Riemannian manifolds, we turn our attention to normal naturally reductive homogeneous spaces in Section 5. The rather simple form of rolling intrinsically pseudo-Riemannian symmetric spaces from [1] motivates an Ansatz, which is an obvious generalization of this rolling. Unfortunately, this does not yield the desired result, in general.…”

Section: Introductionmentioning

confidence: 99%

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Rolling Stiefel Manifolds Equipped with α-Metrics

Schlarb,

Hüper,

Markina

et al. 2023

Mathematics

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We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with α-metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely, by investigating the intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence as to why a seemingly straightforward generalization of the intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained, provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of the rolling Stiefel manifolds known from the literature.

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“…Assume that G/H is a pseudo-Riemannian symmetric space. Then, by [1] (Sec. 4.2), a rolling of p over G/H along a given rolling curve can be viewed as a triple α(t), α(t), A(t) , where…”

Section: No-go Lemmamentioning

confidence: 97%

“…Remark 3. The discussion in [1] (Sec. 3) reveals that a rolling in the sense of Definition 6 is closely related to the classical definition of rolling in [8] (Ap.…”

Section: Definition 6 (Extrinsic Rolling (Ii))mentioning

confidence: 99%

“…Motivated by [1] (Prop. 3), we relate the two different notions of extrinsic rolling from Definitions 5 and 6.…”

Section: Definition 6 (Extrinsic Rolling (Ii))mentioning

confidence: 99%

Section: No-go Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rolling Stiefel Manifolds Equipped with α-Metrics

Schlarb,

Hüper,

Markina

et al. 2023

Mathematics

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Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport

Schlarb

2024

J Geom Anal

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We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.

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Rolling reductive homogeneous spaces

Schlarb

2024

Anal.Math.Phys.

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Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition $$\mathfrak {{g}} = \mathfrak {{h}} \oplus \mathfrak {{m}}$$ g = h ⊕ m , we consider rollings of $$\mathfrak {{m}}$$ m over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle $$\overline{\pi }:\overline{Q}\rightarrow Q$$ π ¯ : Q ¯ → Q over the configuration space equipped with a suitable principal connection, rollings of $$\mathfrak {{m}}$$ m over G/H can be expressed in terms of horizontally lifted curves on $$\overline{Q}$$ Q ¯ . The total space of $$\overline{\pi }:\overline{Q}\rightarrow Q$$ π ¯ : Q ¯ → Q is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of $$\mathfrak {{m}}$$ m over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on $$\overline{Q}$$ Q ¯ , the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples.

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Symmetric Spaces Rolling on Flat Spaces

Cited by 7 publications

References 21 publications

Rolling Stiefel Manifolds Equipped with α-Metrics

Rolling Stiefel Manifolds Equipped with α-Metrics

Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport

Rolling reductive homogeneous spaces

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