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Principal Boundary on Riemannian Manifolds

Zhigang Yao, Zhenyue Zhang
2019 Figshare  
For multivariate datasets lying 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  ... 
doi:10.6084/m9.figshare.8038301.v1 fatcat:vh5rnhpfonfmlf266whkqywutm

Principal Boundary on Riemannian Manifolds

Zhigang Yao, Zhenyue Zhang
2019 Figshare  
For multivariate datasets lying 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  ... 
doi:10.6084/m9.figshare.8038301 fatcat:h7kijelrufesfbxglcpsqallhq

Principal Boundary on Riemannian Manifolds [article]

Zhigang Yao, Zhenyue Zhang
2019 arXiv   pre-print
For multivariate datasets lying 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.  ... 
arXiv:1711.06705v2 fatcat:ha6pat5dxfb5vmvkcedv7ut7ke

On Green's functions for positive, self-adjoint, elliptic pseudo-differential operators on closed, Riemannian manifolds [article]

David Raske
2011 arXiv   pre-print
In this short note we review some facts about elliptic differential operators 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  ... 
arXiv:1003.5848v2 fatcat:ltkkkiivcnarhp25al7ihofb5a

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

Irena Hinterleitner
2013 Publications de l'Institut Mathématique (Beograd)  
We found new criteria for sectional curvatures 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.  ... 
doi:10.2298/pim1308125h fatcat:njewymc2njditlk7blmxbo6d5m

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.  ... 

Page 1169 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
Riemannian manifolds.  ...  principal directions.  ... 

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

Fernando Galaz-Garcia
2012 arXiv   pre-print
We show that any effective isometric torus action of maximal rank 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  ... 
arXiv:1201.1312v1 fatcat:ptkgnrk5ufgh7nqeq5gkz2rdla

A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds [chapter]

Fernando Galaz-Garcia
2014 Lecture notes in mathematics  
We show that any effective isometric torus action of maximal rank 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  ... 
doi:10.1007/978-3-319-06373-7_3 fatcat:3p7wsvby6jbslodcctozvgrupu

On the Weil-Petersson metric on Teichmüller space

A. E. Fischer, A. J. Tromba
1984 Transactions of the American Mathematical Society  
There is a natural 3>0 invariant L2 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  ... 
doi:10.1090/s0002-9947-1984-0742427-x fatcat:dmnywqqfjfa4nemn42fkfpzhiu

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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