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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
All papers appearing in this volume are original research articles
and have not been published elsewhere. They meet the requirements
that are necessary for publication in a good quality primary
journal. E.Belchev, S.Hineva: On the minimal hypersurfaces of a
locally symmetric manifold. -N.Blasic, N.Bokan, P.Gilkey: The
spectral geometry of the Laplacian and the conformal Laplacian for
manifolds with boundary. -J.Bolton, W.M.Oxbury, L.Vrancken, L.M.
Woodward: Minimal immersions of RP2 into CPn. -W.Cieslak, A.
Miernowski, W.Mozgawa: Isoptics of a strictly convex curve.
-F.Dillen, L.Vrancken: Generalized Cayley surfaces. -A.Ferrandez,
O.J.Garay, P.Lucas: On a certain class of conformally flat
Euclidean hypersurfaces. -P.Gauduchon: Self-dual manifolds with
non-negative Ricci operator. -B.Hajduk: On the obstruction group
toexistence of Riemannian metrics of positive scalar curvature.
-U.Hammenstaedt: Compact manifolds with 1/4-pinched negative
curvature. -J.Jost, Xiaowei Peng: The geometry of moduli spaces of
stable vector bundles over Riemannian surfaces. - O.Kowalski,
F.Tricerri: A canonical connection for locally homogeneous
Riemannian manifolds. -M.Kozlowski: Some improper affine spheres in
A3. -R.Kusner: A maximum principle at infinity and the topology of
complete embedded surfaces with constant mean curvature. -Anmin Li:
Affine completeness and Euclidean completeness. -U.Lumiste: On
submanifolds with parallel higher order fundamental form in
Euclidean spaces. -A.Martinez, F.Milan: Convex affine surfaces with
constant affine mean curvature. -M.Min-Oo, E.A.Ruh, P.Tondeur:
Transversal curvature and tautness for Riemannian foliations.
-S.Montiel, A.Ros: Schroedinger operators associated to a
holomorphic map. -D.Motreanu: Generic existence of Morse functions
on infinite dimensional Riemannian manifolds and applications.
-B.Opozda: Some extensions of Radon's theorem.
The problem of finding minimal surfaces, i. e. of finding the
surface of least area among those bounded by a given curve, was one
of the first considered after the foundation of the calculus of
variations, and is one which received a satis factory solution only
in recent years. Called the problem of Plateau, after the blind
physicist who did beautiful experiments with soap films and
bubbles, it has resisted the efforts of many mathematicians for
more than a century. It was only in the thirties that a solution
was given to the problem of Plateau in 3-dimensional Euclidean
space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The
methods of Douglas and Rado were developed and extended in
3-dimensions by several authors, but none of the results was shown
to hold even for minimal hypersurfaces in higher dimension, let
alone surfaces of higher dimension and codimension. It was not
until thirty years later that the problem of Plateau was
successfully attacked in its full generality, by several authors
using measure-theoretic methods; in particular see De Giorgi [DG1,
2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren
[AF1, 2]. Federer and Fleming defined a k-dimensional surface in
IR" as a k-current, i. e. a continuous linear functional on
k-forms. Their method is treated in full detail in the splendid
book of Federer [FH 1].
This book gives an introductory exposition of the theory of
hyperfunctions and regular singularities. This first English
introduction to hyperfunctions brings readers to the forefront of
research in the theory of harmonic analysis on symmetric spaces. A
substantial bibliography is also included. This volume is based on
a paper which was awarded the 1983 University of Copenhagen Gold
Medal Prize.
15 0. PRELIMINARIES a) Notations from Manifold Theory b) The
Language of Jet Manifolds c) Frame Manifolds d) Differentia! Ideals
e) Exterior Differential Systems EULER-LAGRANGE EQUATIONS FOR
DIFFERENTIAL SYSTEMS ~liTH ONE I. 32 INDEPENDENT VARIABLE a)
Setting up the Problem; Classical Examples b) Variational Equations
for Integral Manifolds of Differential Systems c) Differential
Systems in Good Form; the Derived Flag, Cauchy Characteristics, and
Prolongation of Exterior Differential Systems d) Derivation of the
Euler-Lagrange Equations; Examples e) The Euler-Lagrange
Differential System; Non-Degenerate Variational Problems; Examples
FIRST INTEGRALS OF THE EULER-LAGRANGE SYSTEM; NOETHER'S II. 1D7
THEOREM AND EXAMPLES a) First Integrals and Noether's Theorem; Some
Classical Examples; Variational Problems Algebraically Integrable
by Quadratures b) Investigation of the Euler-Lagrange System for
Some Differential-Geometric Variational Pro~lems: 2 i) ( K ds for
Plane Curves; i i) Affine Arclength; 2 iii) f K ds for Space
Curves; and iv) Delauney Problem. II I. EULER EQUATIONS FOR
VARIATIONAL PROBLEfiJS IN HOMOGENEOUS SPACES 161 a) Derivation of
the Equations: i) Motivation; i i) Review of the Classical Case;
iii) the Genera 1 Euler Equations 2 K /2 ds b) Examples: i) the
Euler Equations Associated to f for lEn; but for Curves in i i)
Some Problems as in i) sn; Non- Curves in iii) Euler Equations
Associated to degenerate Ruled Surfaces IV.
Writing this book, I had in my mind areader trying to get some
knowledge of a part of the modern differential geometry. I
concentrate myself on the study of sur faces in the Euclidean
3-space, this being the most natural object for investigation. The
global differential geometry of surfaces in E3 is based on two
classical results: (i) the ovaloids (i.e., closed surfaces with
positive Gauss curvature) with constant Gauss or mean curvature are
the spheres, (u) two isometrie ovaloids are congruent. The results
presented here show vast generalizations of these facts. Up to now,
there is only one book covering this area of research: the Lecture
Notes [3] written in the tensor slang. In my book, I am using the
machinary of E. Cartan's calculus. It should be equivalent to the
tensor calculus; nevertheless, using it I get better results (but,
honestly, sometimes it is too complicated). It may be said that
almost all results are new and belong to myself (the exceptions
being the introductory three chapters, the few classical results
and results of my post graduate student Mr. M. AEFWAT who proved
Theorems V.3.1, V.3.3 and VIII.2.1-6).
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Differential Geometrical Methods in Mathematical Physics
- Proceedings of the Conference Held at Aix-en-Provence, September 3-7, 1979 and Salamanca, September 10-14, 1979
(English, French, Paperback, 1980 ed.)
P.L. Garcia, A. Perez-Rendon, Jean-Marie Souriau
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R1,726
Discovery Miles 17 260
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Ships in 10 - 15 working days
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This volume guides early-career researchers through recent
breakthroughs in mathematics and physics as related to general
relativity. Chapters are based on courses and lectures given at the
July 2019 Domoschool, International Alpine School in Mathematics
and Physics, held in Domodossola, Italy, which was titled "Einstein
Equations: Physical and Mathematical Aspects of General
Relativity". Structured in two parts, the first features four
courses from prominent experts on topics such as local energy in
general relativity, geometry and analysis in black hole spacetimes,
and antimatter gravity. The second part features a variety of
papers based on talks given at the summer school, including topics
like: Quantum ergosphere General relativistic Poynting-Robertson
effect modelling Numerical relativity Length-contraction in curved
spacetime Classicality from an inhomogeneous universe Einstein
Equations: Local Energy, Self-Force, and Fields in General
Relativity will be a valuable resource for students and researchers
in mathematics and physicists interested in exploring how their
disciplines connect to general relativity.
The differential equations which model the action of selection and
recombination are nonlinear equations which are impossible to It is
even difficult to describe in general the solve explicitly.
Recently, Shahshahani began using qualitative behavior of
solutions. differential geometry to study these equations [28].
with this mono graph I hope to show that his ideas illuminate many
aspects of pop ulation genetics. Among these are his proof and
clarification of Fisher's Fundamental Theorem of Natural Selection
and Kimura's Maximum Principle and also the effect of recombination
on entropy. We also discover the relationship between two classic
measures of 2 genetic distance: the x measure and the arc-cosine
measure. There are two large applications. The first is a precise
definition of the biological concept of degree of epistasis which
applies to general (i.e. frequency dependent) forms of selection.
The second is the unexpected appearance of cycling. We show that
cycles can occur in the two-locus-two-allele model of selection
plus recombination even when the fitness numbers are constant (i.e.
no frequency dependence). This work is addressed to two different
kinds of readers which accounts for its mode of organization. For
the biologist, Chapter I contains a description of the entire work
with brief indications of a proof for the harder results. I imagine
a reader with some familiarity with linear algebra and systems of
differential equations. Ideal background is Hirsch and Smale's text
[15].
X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite
des ecrivains au double alphabet, moi qui, il y a plus de quarante
ans deja, avais accueilli sur mes terres un general epris de
mathematiques. JI m'avait parle de ses projets grandioses en
promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je
suis entiche de geometrie et c'est d'elle dontje voudrais vous
parler, oh! certes pas de toute la geometrie, mais de celle que
fait l'artisan qui taille, burine, amene, gauchit, peaufine les
formes. Mon interet pour le probleme dont je veux vous entretenir
ici, je le dois a un ami ebeniste. En effet comme je rendais un
jour visite il cet ami, je le trouvai dans son atelier affaire a un
tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte
de toupie et me dit: "Monsieur Besse, vous qui calculez les formes
avec vos grimoires, que pensez-vous de ceci?)) Je le regardai
interloque. Il poursuivit: "Regardez! Si vous prenez ce collier de
laine et si vous le maintenez fermement avec un doigt place
n'importe ou sur la toupie, eh bien! la toupie passera toujours
juste en son interieur, sans laisser le moindre espace.)) Je
rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une
boule. Je me mis alors au travail ...
The question of existence of c10sed geodesics on a Riemannian
manifold and the properties of the corresponding periodic orbits in
the geodesic flow has been the object of intensive investigations
since the beginning of global differential geo metry during the
last century. The simplest case occurs for c10sed surfaces of
negative curvature. Here, the fundamental group is very large and,
as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed
curve can be deformed into a c10sed curve having minimallength in
its free homotopy c1ass. This minimal curve is, up to the
parameterization, uniquely determined and represents a c10sed
geodesic. The question of existence of a c10sed geodesic on a
simply connected c10sed surface is much more difficult. As pointed
out by Poincare [po 1] in 1905, this problem has much in common
with the problem ofthe existence of periodic orbits in the
restricted three body problem. Poincare [l.c.] outlined a proof
that on an analytic convex surface which does not differ too much
from the standard sphere there always exists at least one c10sed
geodesic of elliptic type, i. e., the corres ponding periodic orbit
in the geodesic flow is infinitesimally stable.
A working knowledge of differential forms so strongly illuminates
the calculus and its developments that it ought not be too long
delayed in the curriculum. On the other hand, the systematic
treatment of differential forms requires an apparatus of topology
and algebra which is heavy for beginning undergraduates. Several
texts on advanced calculus using differential forms have appeared
in recent years. We may cite as representative of the variety of
approaches the books of Fleming [2], (1) Nickerson-Spencer-Steenrod
[3], and Spivak [6]. . Despite their accommodation to the innocence
of their readers, these texts cannot lighten the burden of
apparatus exactly because they offer a more or less full measure of
the truth at some level of generality in a formally precise
exposition. There. is consequently a gap between texts of this type
and the traditional advanced calculus. Recently, on the occasion of
offering a beginning course of advanced calculus, we undertook the
expe- ment of attempting to present the technique of differential
forms with minimal apparatus and very few prerequisites. These
notes are the result of that experiment. Our exposition is intended
to be heuristic and concrete. Roughly speaking, we take a
differential form to be a multi-dimensional integrand, such a thing
being subject to rules making change-of-variable calculations
automatic. The domains of integration (manifolds) are explicitly
given "surfaces" in Euclidean space. The differentiation of forms
(exterior (1) Numbers in brackets refer to the Bibliography at the
end.
This book aims to present to first and second year graduate
students a beautiful and relatively accessible field of
mathematics-the theory of singu larities of stable differentiable
mappings. The study of stable singularities is based on the now
classical theories of Hassler Whitney, who determined the generic
singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~
R2, and Marston Morse, for mappings who studied these singularities
for Rn ~ R. It was Rene Thorn who noticed (in the late '50's) that
all of these results could be incorporated into one theory. The
1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave
the first general exposition of this theory. However, these notes
preceded the work of Bernard Malgrange [23] on what is now known as
the Malgrange Preparation Theorem-which allows the relatively easy
computation of normal forms of stable singularities as well as the
proof of the main theorem in the subject-and the definitive work of
John Mather. More recently, two survey articles have appeared, by
Arnold [4] and Wall [53], which have done much to codify the new
material; still there is no totally accessible description of this
subject for the beginning student. We hope that these notes will
partially fill this gap. In writing this manuscript, we have
repeatedly cribbed from the sources mentioned above-in particular,
the Thom-Levine notes and the six basic papers by Mather.
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