Positively Curved Manifolds With Large Conjugate Radius

Schmidt, Benjamin

doi:10.1142/s1793525313500167

Cited by 4 publications

(5 citation statements)

References 30 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is also reasonable to compare the sectional curvature version of Theorem to the ‘rank rigidity’ results of Schmidt and Shankar–Spatzier–Wilking in and . Shankar, Spatzier, and Wilking obtained the conclusion of Theorem for manifolds with curvature less than or equal to 1 and minimal conjugate radius

π

.…”

Section: Introductionmentioning

confidence: 89%

“…It is also reasonable to compare the sectional curvature version of Theorem 1.2 to the 'rank rigidity' results of Schmidt and Shankar-Spatzier-Wilking in [23] and [25]. Shankar, Spatzier, and Wilking obtained the conclusion of Theorem 1.2 for manifolds with curvature less than or equal to 1 and minimal conjugate radius π. Schmidt proves that if M has sectional curvature 1 and conjugate radius π 2 , then its universal cover is homeomorphic to S n or isometric to a projective space.…”

Section: Introductionmentioning

confidence: 91%

See 1 more Smart Citation

Focal radius, rigidity, and lower curvature bounds

Guijarro

Wilhelm

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point. Using it, we show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed π2. In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective space with their standard metrics, provided that N is closed. Our results also hold for kth intermediate Ricci curvature, provided that the submanifold has dimension ⩾k. Thus, in a manifold with Ricci curvature ⩾n−1, all hypersurfaces have focal radius ⩽π2, and space forms are the only such manifolds where equality can occur, if the submanifold is closed. Example and Remark show that our results cannot be proven using standard Riccati or Jacobi comparison techniques.

show abstract

π

.…”

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 91%

Focal radius, rigidity, and lower curvature bounds

Guijarro

Wilhelm

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…It is also reasonable to compare the sectional curvature version of Theorem B to the "rank rigidity" results of Schmidt and Shankar-Spatzier-Wilking in [24] and [26]. Shankar, Spatzier, and Wilking obtained the conclusion of Theorem B for manifolds with curvature less than or equal to 1 and minimal conjugate radius π. Schmidt proves that if M has sectional curvature ≥ 1 and conjugate radius ≥ π 2 , then its universal cover is homeomorphic to S n or isometric to a projective space.…”

Section: If the Focal Radius Of N Is πmentioning

confidence: 90%

Focal Radius, Rigidity, and Lower Curvature Bounds

Guijarro,

Wilhelm

2016

Preprint

View full text Add to dashboard Cite

We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point.Using it we show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed π 2 . In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective space with their standard metrics, provided N is closed.Our results also hold for k th -intermediate Ricci curvature, provided the submanifold has dimension ≥ k. Thus in a manifold with Ricci curvature ≥ n − 1, all hypersurfaces have focal radius ≤ π 2 , and space forms are the only such manifolds where equality can occur, if the submanifold is closed.Example 2.38 and Remark 3.4 show that our results cannot be proven using standard Riccati or Jacobi comparison techniques.

show abstract

“…In view of Grove-Shiohama's celebrated diameter sphere theorem for positive sectional curvature (see [27]) and the wealth of other sphere theorems for manifolds of positive sectional and of positive Ricci curvature (see, e.g., [4], [31], [14], [37], [2], [8], [18], [22], [26], [29], [33], [34], [36], [39], [42], [44]), it is therefore natural to ask which conditions on the injectivity radius, or, more generally, conjugate radius, of a closed Riemannian n-manifold M with positive scalar curvature will guarantee stability of Green's above-mentioned results in the sense that M can still be recognized as being homeomorphic, or even diffeomorphic, to the standard n-sphere or, respectively, to an n-dimensional spherical space form.…”

Section: Introductionmentioning

confidence: 99%