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University of Notre Dame

1. Xiaoyang Chen. Curvature and Riemannian Submersions</h1>.

Degree: Mathematics, 2014, University of Notre Dame

URL: https://curate.nd.edu/show/fb494744f2q

We study Riemannian submersions from
positively curved manifolds and from Einstein manifolds. We first
prove a diameter rigidity theorem for Riemannian
submersions.Secondly we show that there is no nontrivial Riemannian
submersion from positively curved four manifolds such that either
the mean curvature vector field or the norm of the O'Neill tensor
is basic. We also classify Riemannian submersions from compact
four-dimensional Einstein manifolds with totally geodesic
fibers.
*Advisors/Committee Members: Karsten Grove, Committee Chair, Stephan Stolz, Committee Member, Fred Xavier, Committee Member, Xiaobo Liu, Committee Member.*

Subjects/Keywords: Fred Wilhelm’s conjecture; Riemannian submersions

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chen, X. (2014). Curvature and Riemannian Submersions</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/fb494744f2q

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Chen, Xiaoyang. “Curvature and Riemannian Submersions</h1>.” 2014. Thesis, University of Notre Dame. Accessed December 01, 2020. https://curate.nd.edu/show/fb494744f2q.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Chen, Xiaoyang. “Curvature and Riemannian Submersions</h1>.” 2014. Web. 01 Dec 2020.

Vancouver:

Chen X. Curvature and Riemannian Submersions</h1>. [Internet] [Thesis]. University of Notre Dame; 2014. [cited 2020 Dec 01]. Available from: https://curate.nd.edu/show/fb494744f2q.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chen X. Curvature and Riemannian Submersions</h1>. [Thesis]. University of Notre Dame; 2014. Available from: https://curate.nd.edu/show/fb494744f2q

Not specified: Masters Thesis or Doctoral Dissertation

University of Notre Dame

2. John Harvey. Around Palais' Covering Homotopy Theorem</h1>.

Degree: Mathematics, 2014, University of Notre Dame

URL: https://curate.nd.edu/show/qn59q239w65

The classification by Palais of G-spaces,
topological spaces acted on by homeomorphisms by a compact Lie
group G, is refined. Under mild topological hypotheses, it is shown
that when a sequence of orbit spaces is “close” to a limit orbit
space, in some suitable sense, within a larger ambient orbit space,
the G-spaces in the tail of the sequence are strongly equivalent to
the limit G-space. Three applications of the theory to Alexandrov
and Riemannian geometry are then given. The Covering Homotopy
Theorem, which is key to the classification theory, is used to
prove a version of the Slice Theorem for Alexandrov spaces, showing
that the local action of a group of isometries is topologically
determined by its infinitesimal action. The refinement of the
classification theory is used to prove an equivariant version of
Perelman’s Stability Theorem for equicontinous sequences of
isometric actions by a fixed compact Lie group. The class of
Riemannian orbifolds of a given dimension defined by a lower bound
on the sectional curvature and the volume and an upper bound on the
diameter is shown to be finite up to orbifold homeomorphism.
Furthermore, any class of isospectral Riemannian orbifolds with a
lower bound on the sectional curvature is also shown to be finite
up to orbifold homeomorphism.
*Advisors/Committee Members: Xiaobo Liu, Committee Member, William Dwyer, Committee Member, Stephan Stolz, Committee Member, Karsten Grove, Committee Chair.*

Subjects/Keywords: isometric actions; covering sequence theorem; alexandrov geometry; curvature bounds; isospectral orbifolds; transformation groups

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Harvey, J. (2014). Around Palais' Covering Homotopy Theorem</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/qn59q239w65

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Harvey, John. “Around Palais' Covering Homotopy Theorem</h1>.” 2014. Thesis, University of Notre Dame. Accessed December 01, 2020. https://curate.nd.edu/show/qn59q239w65.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Harvey, John. “Around Palais' Covering Homotopy Theorem</h1>.” 2014. Web. 01 Dec 2020.

Vancouver:

Harvey J. Around Palais' Covering Homotopy Theorem</h1>. [Internet] [Thesis]. University of Notre Dame; 2014. [cited 2020 Dec 01]. Available from: https://curate.nd.edu/show/qn59q239w65.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Harvey J. Around Palais' Covering Homotopy Theorem</h1>. [Thesis]. University of Notre Dame; 2014. Available from: https://curate.nd.edu/show/qn59q239w65

Not specified: Masters Thesis or Doctoral Dissertation

University of Notre Dame

3. Renato G. Bettiol. On different notions of positivity of curvature</h1>.

Degree: Mathematics, 2015, University of Notre Dame

URL: https://curate.nd.edu/show/rn300z72p6p

We study interactions between the geometry
and topology of Riemannian manifolds that satisfy curvature
positivity conditions closely related to positive sectional
curvature (sec>0). First, we discuss two notions of weakly
positive curvature, defined in terms of averages of pairs of
sectional curvatures. The manifold S^{2} x S^{2} is proved to satisfy
these curvature positivity conditions, implying it satisfies a
property intermediate between sec>0 and positive Ricci
curvature, and between sec>0 and nonnegative sectional
curvature. Combined with surgery techniques, this construction
allows to classify (up to homeomorphism) the closed
simply-connected 4-manifolds that admit a Riemannian metric for
which averages of pairs of sectional curvatures of orthogonal
planes are positive. Second, we study the notion of strongly
positive curvature, which is intermediate between sec>0 and
positive-definiteness of the curvature operator. We elaborate on
joint work with Mendes, which yields the classification of
simply-connected homogeneous spaces that admit an invariant metric
with strongly positive curvature. These methods are then used to
study the moduli space of homogeneous metrics with strongly
positive curvature on the Wallach flag manifolds and on Berger
spheres.
*Advisors/Committee Members: Gabor Szekelyhidi, Committee Member, Gerard K. Misiolek, Committee Member, Karsten Grove, Committee Chair, Frederico J. Xavier, Committee Member.*

Subjects/Keywords: Curvature; 4-manifolds; Riemannian geometry; Sectional curvature; Homogeneous spaces; Curvature operator; Metric deformations

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Bettiol, R. G. (2015). On different notions of positivity of curvature</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/rn300z72p6p

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Bettiol, Renato G.. “On different notions of positivity of curvature</h1>.” 2015. Thesis, University of Notre Dame. Accessed December 01, 2020. https://curate.nd.edu/show/rn300z72p6p.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Bettiol, Renato G.. “On different notions of positivity of curvature</h1>.” 2015. Web. 01 Dec 2020.

Vancouver:

Bettiol RG. On different notions of positivity of curvature</h1>. [Internet] [Thesis]. University of Notre Dame; 2015. [cited 2020 Dec 01]. Available from: https://curate.nd.edu/show/rn300z72p6p.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bettiol RG. On different notions of positivity of curvature</h1>. [Thesis]. University of Notre Dame; 2015. Available from: https://curate.nd.edu/show/rn300z72p6p

Not specified: Masters Thesis or Doctoral Dissertation