Normal bundle

Normal vectors on a surface in three-dimensional space

The normal bundle is a term from differential topology and differential geometry , sub-areas of mathematics . Such a vector bundle includes all normal vectors of a submanifold and is therefore a concept that is complementary to the tangential bundle .

With the help of normal bundles, for example, tubular environments of submanifolds can be constructed.

definition

Submanifold

The normal bundle of a differentiable submanifold is the vector bundle over , which consists of all pairs , where applies and is a vector in the quotient space , where and are the tangent spaces of and . In other words, the normal bundle is defined as the disjoint union ${\displaystyle Y\subset X}$ ${\displaystyle Y}$ ${\displaystyle (y,\nu )}$ ${\displaystyle y\in Y}$ ${\displaystyle \nu }$ ${\displaystyle T_{y}X/T_{y}Y}$ ${\displaystyle T_{y}X}$ ${\displaystyle T_{x}Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle NY:=\coprod _{y\in Y}T_{y}X/T_{y}Y}$.

Immersed submanifold

Somewhat more general is the construction of the normal bundle of an immersed submanifold . So be an immersion of in . Then the normal bundle of is defined by ${\displaystyle f\colon Y\to X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle NY:=\coprod _{y\in Y}f^{*}(T_{y}X)/T_{y}Y}$,

where the return is from . ${\displaystyle f^{*}}$ ${\displaystyle f}$

Riemannian geometry

Let and be Riemannian manifolds and be an immersion such that there is a manifold immersed in. Let and be the tangent space of in . Due to the Riemannian metric, there is an orthogonal decomposition of this tangent space. It is the normal space at the point . The amount ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon Y\to X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle p\in Y}$ ${\displaystyle T_{p}X}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle T_{p}X=T_{p}Y\oplus N_{p}Y}$ ${\displaystyle N_{p}Y}$ ${\displaystyle p}$

${\displaystyle NY:=\coprod _{p\in Y}N_{p}Y}$

is the normal bundle of the Riemannian manifold with respect to . This normal bundle in Riemannian geometry is a special case of the definition mentioned above, because it is obviously isomorphic to the quotient spaces of the above definition. ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle N_{p}Y}$

Stable normal bundle

→ Main article : Stable normal bundle

Abstract differentiable manifolds have a canonical tangent bundle but no normal bundle. Only embedding (or immersing) one manifold in another gives a normal bundle.

However, since every differentiable manifold can be embedded in according to Whitney's embedding theorem , every manifold allows a normal bundle in such an embedding. In general there is no natural choice of embedding, but for a given manifold any two embeddings in are isotopic for sufficiently large and therefore induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle as it can vary) is called the stable normal bundle . ${\displaystyle \mathbb {R} ^{N}}$ ${\displaystyle \mathbb {R} ^{N}}$ ${\displaystyle N}$ ${\displaystyle N}$

Web links

• Normal bundle . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ^ ^{A b} Antoni A. Kosinski: Differential Manifolds . Academic Press Limited, New Brunswick, New Jersey 1992, ISBN 0-12-421850-4 , pp. 44 .
2. ↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , pp. 132-133.