A universal Riemannian foliated space

López, Jesús A. Álvarez; Lijó, Ramón Barral; Candel, Alberto

doi:10.1016/j.topol.2015.11.006

Cited by 10 publications

(26 citation statements)

References 16 publications

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“…Standard references about foliated spaces are , [, Chapter 11], [, Part 1] and . See also [, Section 2.1] for a quick summary of what is needed here.…”

Section: Introductionmentioning

confidence: 99%

“…It is commonly accepted that such a result should be true, and that it should follow by using the closure of the canonical embedding of the manifold into the Gromov space

M_{*}

of pointed proper metric spaces , [, Chapter 3], or, better, into its smooth version, the space

{scriptM}_{*}^{\infty} false(n false)

of isometry classes of pointed complete connected Riemannian n ‐manifolds with the topology defined by the

C^{\infty}

convergence [, Chapter 10, Section 3.2], [, Theorem 1.2]. However, to the authors knowledge, no complete proof has been given so far.…”

Section: Introductionmentioning

confidence: 99%

“…The restriction of

{scriptF}_{*} false(n false)

{scriptM}_{*, lnp}^{\infty} false(n false)

is denoted by

{scriptF}_{*, lnp} false(n false)

. For

n \geq 2

{scriptM}_{*, lnp}^{\infty} false(n false)

is open and dense in

{scriptM}_{*}^{\infty} false(n false)

, and

{scriptF}_{*, lnp} false(n false)

is a Riemannian foliated space of dimension n so that each map

ι_{M} : M \to im ι_{M}

is a local isometry and the holonomy covering of the leaf

im ι_{M}

[, Theorem 1.3]; in particular,

{scriptM}_{*, np}^{\infty} false(n false)

is the union of leaves with trivial holonomy. Moreover

{prefixCl}_{\infty} false(prefixim ι_{M} false)

is compact if and only if M is of bounded geometry [, Theorem 12.3] (see also , [, Chapter 10, Sections and 4]), where

{Cl}_{\infty}

denotes the closure operator in

{scriptM}_{*}^{\infty} false(n false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover

{prefixCl}_{\infty} false(prefixim ι_{M} false)

is compact if and only if M is of bounded geometry [, Theorem 12.3] (see also , [, Chapter 10, Sections and 4]), where

{Cl}_{\infty}

denotes the closure operator in

{scriptM}_{*}^{\infty} false(n false)

. Then, analyzing the cases where

{prefixCl}_{\infty} false(prefixim ι_{M} false) \subset {scriptM}_{*, lnp} false(n false)

, a version of Theorem follows assuming restrictions on M [, Theorem 1.5].…”

Section: Introductionmentioning

confidence: 99%

“…It is possible to give a version of Theorem closer to [, Theorem 1.3], using the subspace

{\overset{M}{true}}_{*, lnp}^{\infty} false(n false)

consisting of the classes

[M, f, x]

such that

M \to Iso (M, f) ∖ M

is a covering map. Such a result could be proved with the obvious adaptation of the proof of [, Theorem 1.3], using the exponential map to define foliated charts. Instead, we have opted for studying

{\overset{M}{true}}_{*, imm}^{\infty} false(n false)

because, in this case, the immersions f directly provide foliated charts.…”

Section: Introductionmentioning

confidence: 99%

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Bounded geometry and leaves

López

Lijó

2017

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“…Standard references about foliated spaces are , [, Chapter 11], [, Part 1] and . See also [, Section 2.1] for a quick summary of what is needed here.…”

Section: Introductionmentioning

confidence: 99%

“…It is commonly accepted that such a result should be true, and that it should follow by using the closure of the canonical embedding of the manifold into the Gromov space

M_{*}

of pointed proper metric spaces , [, Chapter 3], or, better, into its smooth version, the space

{scriptM}_{*}^{\infty} false(n false)

of isometry classes of pointed complete connected Riemannian n ‐manifolds with the topology defined by the

C^{\infty}

convergence [, Chapter 10, Section 3.2], [, Theorem 1.2]. However, to the authors knowledge, no complete proof has been given so far.…”

Section: Introductionmentioning

confidence: 99%

“…The restriction of

{scriptF}_{*} false(n false)

{scriptM}_{*, lnp}^{\infty} false(n false)

is denoted by

{scriptF}_{*, lnp} false(n false)

. For

n \geq 2

{scriptM}_{*, lnp}^{\infty} false(n false)

is open and dense in

{scriptM}_{*}^{\infty} false(n false)

, and

{scriptF}_{*, lnp} false(n false)

is a Riemannian foliated space of dimension n so that each map

ι_{M} : M \to im ι_{M}

is a local isometry and the holonomy covering of the leaf

im ι_{M}

[, Theorem 1.3]; in particular,

{scriptM}_{*, np}^{\infty} false(n false)

is the union of leaves with trivial holonomy. Moreover

{prefixCl}_{\infty} false(prefixim ι_{M} false)

is compact if and only if M is of bounded geometry [, Theorem 12.3] (see also , [, Chapter 10, Sections and 4]), where

{Cl}_{\infty}

denotes the closure operator in

{scriptM}_{*}^{\infty} false(n false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover

{prefixCl}_{\infty} false(prefixim ι_{M} false)

is compact if and only if M is of bounded geometry [, Theorem 12.3] (see also , [, Chapter 10, Sections and 4]), where

{Cl}_{\infty}

denotes the closure operator in

{scriptM}_{*}^{\infty} false(n false)

. Then, analyzing the cases where

{prefixCl}_{\infty} false(prefixim ι_{M} false) \subset {scriptM}_{*, lnp} false(n false)

, a version of Theorem follows assuming restrictions on M [, Theorem 1.5].…”

Section: Introductionmentioning

confidence: 99%

“…It is possible to give a version of Theorem closer to [, Theorem 1.3], using the subspace

{\overset{M}{true}}_{*, lnp}^{\infty} false(n false)

consisting of the classes

[M, f, x]

such that

M \to Iso (M, f) ∖ M

is a covering map. Such a result could be proved with the obvious adaptation of the proof of [, Theorem 1.3], using the exponential map to define foliated charts. Instead, we have opted for studying