This book is devoted to the rst acquaintance with the dierential geometry
Therefore it begins with the theory of curves in three-dimensional Euclidean spac
E. Then the vectorial analysis in E is stated both in Cartesian and curvilinea
coordinates, afterward the theory of surfaces in the space E is considered.
The newly fashionable approach starting with the concept of a dierentiabl
manifold, to my opinion, is not suitable for the introduction to the subject.
The Ricci ﬂow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. ...
John Archibald Wheeler was born on July 9, 1911, in Jacksonville, Florida, and
passed away on April 13, 2008, in Hightstown, New Jersey; his influence on gravitational
physics and science in general will remain forever. Among his many and
important contributions to physics, he was one of the fathers of the renaissance of
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). ...
We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a “bolic”, weakly geodesic metric space of bounded geometry. 1. Introduction This work has grown out of an attempt to give a purely KK-theoretic proof of a result of A. Connes and H. Moscovici ([CM], [CGM]) that hyperbolic groups satisfy the Novikov conjecture. ...
The space of embedded minimal surfaces of ﬁxed genus in a 3-manifold II; Multi-valued graphs in disks
By Tobias H. Colding and William P. Minicozzi II*
0. Introduction This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 . We show here that if the curvature of such a disk...
This book is based on lectures delivered over the years by the author at the
Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introductions
to the basic theorems of differential geometry and to elasticity theory in
The treatment is essentially self-contained and proofs are complete.
Next, a few words about our strategy. It is well recognized now that one has to go beyond
the Einstein-Hilbert action for gravity, both from the experimental viewpoint (eg.,because of
Dark Energy) and from the theoretical viewpoint (eg., because of the UV incompleteness
of quantized Einstein gravity, and the need of its uniﬁcation with the Standard Model of
In our approach, the origin of inﬂation is purely geometrical, ie. is closely related to
space-time and gravity.
Invariant measures for the geodesic ﬂow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and well-studied subject. This paper continues an investigation into a 2-dimensional analog of this ﬂow for a 3-manifold N . Namely, the article discusses 2-dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1, surfaces which are called k-surfaces.
There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diﬀeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold (M, g) is up to homotopy given by a rank one symmetric space, provided that its isometry group Iso(M, g) is large. More precisely we prove ﬁrst that if dim(Iso(M, g)) ≥ 2 dim(M ) − 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M...
Chúng tôi nghiên cứu đóng submanifolds M kích thước 2n + 1, đắm mình vào một (4N + 1) chiều Sasakian hình thức không gian (N, ξ, η, φ) với c cong φ-cắt liên tục, như vậy mà lĩnh vực vector Reeb ξ là tiếp tuyếnM.
Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds: An Introduction to Relativity 8.
This work develops the geometry and dynamics of mechanical systems with
nonholonomic constraints and symmetry from the perspective of Lagrangian me-
chanics and with a view to control-theoretical applications. The basic methodology
is that of geometric mechanics applied to the Lagrange-d’Alembert formulation,
generalizing the use of connections and momentum maps associated with a given
symmetry group to this case.