zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nilpotent structures and invariant metrics on collapsed manifolds. (English) Zbl 0758.53022
Let $M$ be a complete Riemannian $n$-manifold of bounded curvature. For any $\varepsilon > 0$, the $\varepsilon$-collapsed part of $M$ is defined as the set ${\cal C}\sp n(\varepsilon)$ of points at which the injectivity radius of the exponential map is $<\varepsilon$. The authors study the structure of the $\varepsilon$-collapsed part of a manifold $M$ for suitably small $\varepsilon$. The main results show that the local geometry of ${\cal C}\sp n(\varepsilon)$ is encoded partially in the symmetry properties of a nearby metric. More precisely, a given metric can be closely approximated by one that admits a sheaf of nilpotent Lie algebras of local Killing vector fields pointing in all sufficiently collapsed directions of ${\cal C}\sp n(\varepsilon)$. This sheaf is called the nilpotent Killing structure. A detailed construction of this nilpotent Killing structure is presented, its properties are established and some applications to the description of collapses of a Riemannian manifold $M$ are indicated.

53C20Global Riemannian geometry, including pinching
Full Text: DOI