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## Notes on projective differential geometry

Fonte: Springer
Publicador: Springer

Tipo: Parte de Livro

Publicado em //2008
EN

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Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ‘Bernstein-Gelfand-Gelfand machinery’. As such, it is completely parallel to conformal differential geometry. On the other hand, there are direct applications within Riemannian differential geometry. We shall soon see, for example, a good geometric reason why the symmetries of the Riemann curvature tensor constitute an irreducible representation of SL(n,ℝ) (rather than SO(n) as one might naively expect). Projective differential geometry also provides the simplest setting in which overdetermined systems of partial differential equations naturally arise.; http://www.springer.com/math/dyn.+systems/book/978-0-387-73830-7; Michael Eastwood

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## Extended Derdzinski-Shen theorem for the Riemann tensor

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/01/2011

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We extend a classical result by Derdzinski and Shen, on the restrictions
imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor.
The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms)
as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"),
typical of some well known differential structures.; Comment: 5 pages

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## Conformally Osserman manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/10/2008

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An algebraic curvature tensor is called Osserman if the eigenvalues of the
associated Jacobi operator are constant on the unit sphere. A Riemannian
manifold is called conformally Osserman if its Weyl conformal curvature tensor
at every point is Osserman. We prove that a conformally Osserman manifold of
dimension $n \ne 3, 4, 16$ is locally conformally equivalent either to a
Euclidean space or to a rank-one symmetric space.; Comment: 23 pages

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## Weyl compatible tensors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematical Physics#General Relativity and Quantum Cosmology#53B20, 53C50 (primary), 83C20 (secondary)

We introduce the new algebraic property of Weyl compatibility for symmetric
tensors and vectors. It is strictly related to Riemann compatibility, which
generalizes the Codazzi condition while preserving much of its geometric
implications. In particular it is shown that the existence of a Weyl compatible
vector implies the Weyl tensor to be algebraically special, and it is a
necessary and sufficient condition for the magnetic part to vanish. Some
theorems (Derdzinski and Shen, Hall) are extended to the broader hypothesis of
Weyl or Riemann compatibility. Weyl compatibility includes conditions that were
investigated in the literature of general relativity (as McIntosh et al.).
Hypersurfaces of pseudo Euclidean spaces provide a simple example of Weyl
compatible Ricci tensor.; Comment: 14 pages. References and comments added

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## Geometric Realizations of Hermitian curvature models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2008

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We show that a Hermitian algebraic curvature model satisfies the Gray
identity if and only if it is geometrically realizable by a Hermitian manifold.
Furthermore, such a curvature model can in fact be realized by a Hermitian
manifold of constant scalar curvature and constant *-scalar curvature which
satisfies the Kaehler condition at the point in question.

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## Geometry of CR submanifolds of maximal CR dimension in complex space forms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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On real hypersurfaces in complex space forms many results are proven. In this
paper we generalize some results concerning extrinsic geometry of real
hypersurfaces, to CR submanifolds of maximal CR dimension in complex space
forms.; Comment: 8 pages

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## Non-existence of CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/12/2010

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It has been proved that there are no real hypersurfaces satisfying RA = 0 in
non-flat complex space forms. In this paper we prove that the same is true in
the case of CR submanifolds of maximal CR dimension, that is there are no CR
submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex
space forms.; Comment: 8 pages

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## R-separation of variables for the conformally invariant Laplace equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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The conditions for R-separation of variables for the conformally invariant
Laplace equation on an n-dimensional Riemannian manifold are determined and
compared with the conditions for the additive separation of the null geodesic
Hamilton-Jacobi equation. The case of 3-dimensions is examined in detail and it
is proven that on any conformally flat manifold the two equations separate in
the same coordinates.; Comment: 13 pages, submitted to Journal of Geometry and Physics. Replaced due
to a factor of 1/4 error found in some presented formulae; note this does not
affect the results derived and discussed - only in the initial presentation

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## Group-theoretic Description of Riemannian Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/04/2007

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It is shown that a locally geometrical structure of arbitrarily curved
Riemannian space is defined by a deformed group of its diffeomorphisms; Comment: 14 pages

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## Canonical Deformed Groups of Diffeomorphisms and Finite Parallel Transports in Riemannian Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We show that finite parallel transports of vectors in Riemannian spaces,
determined by the multiplication law in the deformed groups of diffeomorphisms,
and sequences of infinitesimal parallel transports of vectors along geodesics
are equivalent.; Comment: 12 pages

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## Geometric Realizations of para-Hermitian curvature models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/02/2009

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We show that a para-Hermitian algebraic curvature model satisfies the
para-Gray identity if and only if it is geometrically realizable by a
para-Hermitian manifold. This requires extending the Tricerri-Vanhecke
curvature decomposition to the para-Hermitian setting. Additionally, the
geometric realization can be chosen to have constant scalar curvature and
constant *-scalar curvature.

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## Transforms for minimal surfaces in the 5-sphere

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/02/2005

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We define two transforms between minimal surfaces with non-circular ellipse
of curvature in the 5-sphere, and show how this enables us to construct, from
one such surface, a sequence of such surfaces. We also use the transforms to
show how to associate to such a surface a corresponding ruled minimal
Lagrangian submanifold of complex projective 3-space. We illustrate this
explicitly in the case of bipolar minimal surfaces.

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## Powers of the space forms curvature operator and geodesics of the tangent bundle

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2005

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It is well-known that if a curve is a geodesic line of the tangent (sphere)
bundle with Sasaki metric of a locally symmetric Riemannian manifold then the
projected curve has all its geodesic curvatures constant. In this paper we
consider the case of tangent (sphere) bundle over the real, complex and
quaternionic space form and give a unified proof of the following property: all
geodesic curvatures of projected curve are zero starting from k_3,k_6 and
k_{10} for the real, complex and quaternionic space formes respectively.; Comment: 14 pages

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## Inhomogeneous Ambient Metrics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/11/2006

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An extension of the ambient metric construction of Fefferman-Graham to
infinite order in even dimensions is described. The main ingredients are the
introduction of "inhomogeneous ambient metrics" with asymptotic expansions
involving the logarithm of a defining function homogeneous of degree 2, and an
invariant procedure for taking the smooth part of such an inhomogeneous ambient
metric. The metrics which result depend on the choice of an "ambiguity tensor"
as well as a conformal class. An application to the description of scalar
conformal invariants in even dimensions is outlined.; Comment: 18 pages

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## Regularization by sup-inf convolutions on Riemannian manifolds: an extension of Lasry-Lions theorem to manifolds of bounded curvature

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/01/2014

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#Mathematics - Differential Geometry#Mathematics - Analysis of PDEs#Mathematics - Functional Analysis#53C21, 58B10, 46T05, 35F21, 58C20, 53B20

We show how Lasry-Lions's result on regularization of functions defined on
$\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of
distances can be extended to (finite or infinite dimensional) Riemannian
manifolds $M$ of bounded sectional curvature. More specifically, among other
things we show that if the sectional curvature $K$ of $M$ satisfies $-K_0\leq
K\leq K_0$ on $M$ for some $K_0>0$, and if the injectivity and convexity radii
of $M$ are strictly positive, then every bounded, uniformly continuous function
$f:M\to\mathbb{R}$ can be uniformly approximated by globally $C^{1,1}$
functions defined by $$ (f_{\lambda})^{\mu}=\sup_{z\in M}\inf_{y\in
M}\{f(y)+\frac{1}{2\lambda} d(z,y)^{2}-\frac{1}{2\mu}d(x,z)^2\} $$ as $\lambda,
\mu\to 0^{+}$, with $0<\mu<\lambda/2$. Our definition of (global) $C^{1,1}$
smoothness is intrinsic and natural, and it reduces to the usual one in flat
spaces, but we warn the reader that, in the noncompact case, this definition
differs from other notions of (rather local) $C^{1,1}$ smoothness that have
been recently used, for instance, by A. Fathi and P. Bernard (based on charts).
The importance of this regularization method lies (rather than on the degree of
smoothness obtained) on the fact that the correspondence $f\mapsto
(f_{\lambda})^{\mu}$ is explicit and preserves many significant geometrical
properties that the given functions $f$ may have...

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## Remarks on Bianchi sums and Pontrjagin classes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/02/2014

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We use the exterior and composition products of double forms together with
the alternating operator to reformulate Pontrjagin classes and all Pontrjagin
numbers in terms of the Riemannian curvature. We show that the alternating
operator is obtained by a succession of applications of the first Bianchi sum
and we prove some useful identities relating the previous four operations on
double forms. As an application, we prove that for a $k$-conformally flat
manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any
$i\geq k$. Finally, we study the equality case in an inequality of Thorpe
between the Euler-Poincar\'e charateristic and the $k$-th Pontrjagin number of
a $4k$-dimensional Thorpe manifold.; Comment: 19 pages

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## Interactions between the composition and exterior products of double forms and applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/02/2014

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We translate into the double forms formalism the basic identities of Greub
and Greub-Vanstone that were obtained in the mixed exterior algebra. In
particular, we introduce a second product in the space of double forms, namely
the composition product, which provides this space with a second associative
algebra structure. The composition product interacts with the exterior product
of double forms; the resulting relations provide simple alternative proofs to
some classical linear algebra identities as well as to recent results in the
exterior algebra of double forms.\\ We define a refinement of the notion of
pure curvature of Maillot and we use one of the basic identities to prove that
if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its
Pontrjagin class of degree $4k$ vanishes.; Comment: 24 pages

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## Levi-Civita's Theorem for Noncommutative Tori

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Operator Algebras#Mathematics - Differential Geometry#46L87 (Primary), 58B34, 46L08, 46L08, 53B20 (Secondary)

We show how to define Riemannian metrics and connections on a noncommutative
torus in such a way that an analogue of Levi-Civita's theorem on the existence
and uniqueness of a Riemannian connection holds. The major novelty is that we
need to use two different notions of noncommutative vector field. Levi-Civita's
theorem makes it possible to define Riemannian curvature using the usual
formulas.; Comment: Proposition 3.4 corrected

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## Weakly Z symmetric manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/02/2011

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We introduce a new kind of Riemannian manifold that includes weakly-, pseudo-
and pseudo projective- Ricci symmetric manifolds. The manifold is defined
through a generalization of the so called Z tensor; it is named "weakly Z
symmetric" and denoted by (WZS)_n. If the Z tensor is singular we give
conditions for the existence of a proper concircular vector. For non singular Z
tensor, we study the closedness property of the associated covectors and give
sufficient conditions for the existence of a proper concircular vector in the
conformally harmonic case, and the general form of the Ricci tensor. For
conformally flat (WZS)_n manifolds, we derive the local form of the metric
tensor.; Comment: 13 pages

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## Torsion and the second fundamental form for distributions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/06/2015

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The second fundamental form of Riemannian geometry is generalised to the case
of a manifold with a linear connection and an integrable distribution. This
bilinear form is generally not symmetric and its skew part is the torsion. The
form itself is closely related to the shape map of the connection. The
codimension one case generalises the traditional shape operator of Riemannian
geometry.

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