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Principal Boundary on Riemannian Manifolds

2019
*
Figshare
*

For multivariate datasets lying

doi:10.6084/m9.figshare.8038301.v1
fatcat:vh5rnhpfonfmlf266whkqywutm
*on*an embedded nonlinear*Riemannian**manifold*within the higher-dimensional ambient space, we aim to acquire a classification*boundary*for the classes with labels, using ... the intrinsic metric*on*the*manifolds*. ... Extension to co-dimension*one**principal**boundary*If the*principal*sub-*manifolds*γ 1 , γ 2 of dimension m − 1 (m > 2), are estimated from two sets of samples, we have finite projected points of these samples ...##
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Principal Boundary on Riemannian Manifolds

2019
*
Figshare
*

For multivariate datasets lying

doi:10.6084/m9.figshare.8038301
fatcat:h7kijelrufesfbxglcpsqallhq
*on*an embedded nonlinear*Riemannian**manifold*within the higher-dimensional ambient space, we aim to acquire a classification*boundary*for the classes with labels, using ... the intrinsic metric*on*the*manifolds*. ... Extension to co-dimension*one**principal**boundary*If the*principal*sub-*manifolds*γ 1 , γ 2 of dimension m − 1 (m > 2), are estimated from two sets of samples, we have finite projected points of these samples ...##
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Principal Boundary on Riemannian Manifolds
[article]

2019
*
arXiv
*
pre-print

For multivariate datasets lying

arXiv:1711.06705v2
fatcat:ha6pat5dxfb5vmvkcedv7ut7ke
*on*an embedded nonlinear*Riemannian**manifold*within the higher-dimensional ambient space, we aim to acquire a classification*boundary*for the classes with labels, using ... the intrinsic metric*on*the*manifolds*. ... We thank Zengyan Fan and Wee Chin Tan for reading our manuscript and providing helpful comments*on*the manuscript. ...##
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On Green's functions for positive, self-adjoint, elliptic pseudo-differential operators on closed, Riemannian manifolds
[article]

2011
*
arXiv
*
pre-print

In this short note we review some facts about elliptic differential operators

arXiv:1003.5848v2
fatcat:ltkkkiivcnarhp25al7ihofb5a
*on**Riemannian**manifolds*. ...*on**manifolds*. ... Most of these results are well known for elliptic operators defined*on*open subset of R ⋉ , with Dirichlet*boundary*conditions enforced, but there is little literature*on*what is the case for*manifolds*...##
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Page 402 of Mathematical Reviews Vol. 37, Issue 2
[page]

1969
*
Mathematical Reviews
*

This is a well organized and carefully written exposition of
the results, up to 1966,

*on*8-pinched*Riemannian*and Kahlerian*manifolds*based*on*geodesics and Morse theory. ... The mean curvature H(z,, ---, z,) of the surface is given*on*& region 2. homeomorphic to a ball. The edge of the surface is given by a function A(zx,, ---, z,)*on*the*boundary*@Q of ...##
###
Page 416 of Mathematical Reviews Vol. , Issue 99a
[page]

1991
*
Mathematical Reviews
*

Two compact

*Riemannian**manifolds*are called isospectral if the associated Laplace operators acting*on*functions (with appropriate*boundary*conditions in the case of*manifolds*with*boundary*) have the same ... Let (¥,g) be a*Riemannian**manifold*with positive Ricci curvature and let P be a*principal*bundle over Y with compact structure group. ...##
###
Page 3715 of Mathematical Reviews Vol. , Issue 87g
[page]

1987
*
Mathematical Reviews
*

space of generic connec- tions

*on*a*principal*fibre bundle and the Nijenhuis tensor of the natural almost complex structure*on*the space of almost complex structures*on*a*manifold*. ... The smooth*Riemannian**manifold*M is said to have an isoperi- metric function f if, for every open subset D C M of finite volume v and with smooth*boundary*, the area of the*boundary*OD is not less than ...##
###
Page 287 of Mathematical Reviews Vol. 32, Issue 2
[page]

1966
*
Mathematical Reviews
*

Mat.
17 (1957/58), 95-131; MR 21 #3011; ibid. 21 (1961/62),
127-129; and “Vector fields and infinitesimal trans-
formations

*on**Riemannian**manifolds*with*boundary*’”’, to appear]*on*harmonic differential ... forms defined*on**manifolds*with*boundary*; but now various types of covariant ten- sors (vector fields) are discussed*on*such*manifolds*. ...##
###
Geodesic mappings on compact riemannian manifolds with conditions on sectional curvature

2013
*
Publications de l'Institut Mathématique (Beograd)
*

We found new criteria for sectional curvatures

doi:10.2298/pim1308125h
fatcat:njewymc2njditlk7blmxbo6d5m
*on*compact*Riemannian**manifolds*for which geodesic mappings are affine, and, moreover, homothetic. 2010 Mathematics Subject Classification: Primary 53C20; ... Theorem 5. 2 . 2 Assume a compact*Riemannian**manifold*(M, g) without*boundary*of dimension n 2. ... . , e n } in which Theorem 5 . 1 . 51 Assume a compact*Riemannian**manifold*(M, g) without*boundary*of dimension n 2. ...##
###
Page 476 of Mathematical Reviews Vol. , Issue 97A
[page]

1997
*
Mathematical Reviews
*

Let M be a semi-

*Riemannian**manifold*with (possibly empty)*boundary*. If the metric*on*M is indefinite then the Laplace- Beltrami equation Af = 0*on*M is an ultra-hyperbolic type equa- tion. ... The main theorem states: Let (M,g) be a compact C™*Riemannian**manifold*with piecewise smooth*boundary*, and let {y,;} be an orthonormal set of eigenfunctions for the Laplacian*on*M. ...##
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Page 1169 of Mathematical Reviews Vol. , Issue 2002B
[page]

2002
*
Mathematical Reviews
*

*Riemannian*

*manifolds*. ...

*principal*directions. ...

##
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A note on maximal symmetry rank, quasipositive curvature, and low dimensional manifolds
[article]

2012
*
arXiv
*
pre-print

We show that any effective isometric torus action of maximal rank

arXiv:1201.1312v1
fatcat:ptkgnrk5ufgh7nqeq5gkz2rdla
*on*a compact*Riemannian**manifold*with positive (sectional) curvature and maximal symmetry rank, that is,*on*a positively curved sphere, ... We show that a compact, simply connected*Riemannian*4- or 5-*manifold*of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere ... of directions at an orbit with isotropy T 2 in M * Since (M n , g) is a quasipositively curved*Riemannian**manifold*, M * is a nonnegatively curved 2-*manifold*with non-smooth*boundary*and positive curvature ...##
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A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds
[chapter]

2014
*
Lecture notes in mathematics
*

We show that any effective isometric torus action of maximal rank

doi:10.1007/978-3-319-06373-7_3
fatcat:3p7wsvby6jbslodcctozvgrupu
*on*a compact*Riemannian**manifold*with positive (sectional) curvature and maximal symmetry rank, that is,*on*a positively curved sphere, ... We show that a compact, simply connected*Riemannian*4or 5-*manifold*of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere. ... of directions at an orbit with isotropy T 2 in M * Since (M n , g) is a quasipositively curved*Riemannian**manifold*, M * is a nonnegatively curved 2-*manifold*with non-smooth*boundary*and positive curvature ...##
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On the Weil-Petersson metric on Teichmüller space

1984
*
Transactions of the American Mathematical Society
*

There is a natural 3>0 invariant L2

doi:10.1090/s0002-9947-1984-0742427-x
fatcat:dmnywqqfjfa4nemn42fkfpzhiu
*Riemannian*structure*on*si which induces a*Riemannian*structure*on*si/3>n. ... Teichmüller space for a compact oriented surface M without*boundary*is described as the quotient jí/B0, where s? ... in [2, p. 186], this is a scalar multiple of the Weil-Petersson metric after identifying, in another fashion, the tangent space of Teichmüller space as the space of holomorphic quadratic differentials*on*...##
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Page 6778 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

Let M be a

*Riemannian**manifold*with*boundary*. ... is a*principal*S'-bundle, using a connection w*one*defines a*Riemannian*structure g*on*P such that f is*Riemannian*and geodesic, the induced structure*on*the fibers being the standard*one*. ...
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