Tensor analysis

The tensor or tensor analysis is a branch of differential geometry or the differential topology . It generalizes vector analysis . For example, the differential operator rotation can be generalized to n dimensions in this context. The central objects of tensor analysis are tensor fields . It is examined how differential operators work on these fields.

overview

The tensor calculus was developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita in particular at the beginning of the 20th century , and the central objects of this calculus were the tensors. From this tensor calculus, which is also called Ricci calculus, today's tensor analysis emerged, which is a sub-area of differential geometry.

Through Albert Einstein , for whose theory of relativity the tensor calculus was fundamental, the calculus became well known. The objects that were called tensors at the time are now called tensor fields and are examined for their analytical properties in tensor analysis . Imprecise and formulated in modern terminology are functions that assign a tensor to each point .

In this case, tensor means a purely algebraic object. The concept of the tensor has changed over the course of time, but even today one speaks mostly (but imprecisely) of tensors for tensor fields. However, since only tensor fields and not “correct” tensors are considered in the area of differential geometry or tensor analysis, there is little risk of confusion when creating these terms.

As already mentioned, tensor fields are examined for their analytical properties; in particular, it is possible to derive or differentiate them in a certain way. It is investigated which properties the corresponding differential operators have and how the tensor fields behave with regard to differentiation. In particular, a tensor field is obtained again by differentiating a tensor field. In order to be able to define these important tensor fields at all, the tensor bundle must first be explained. This is a particular vector bundle that is precisely defined in the Tensor Bundle section . Tensor fields are then particularly smooth images that map into this vector bundle.

In tensor analysis, the behavior of geometric differential operators on tensor fields is examined. An important example for a differential operator is the outer derivative on the differential forms , because the differential forms are special tensor fields. The outer derivative can be understood as a generalization of the total differential (for differential forms). With their help, the differential operators known from vector analysis can be generalized. The tensor fields themselves also receive a generalization in tensor analysis: the tensor densities . With their help, coordinate transformations can be carried out in curved spaces, the manifolds .

Central definitions

Tensor bundle

The (r, s) -tensor bundle is a vector bundle whose fibers are (r, s) -tensor spaces over a vector space . So be a differentiable manifold and the tangential bundle with the fibers at the point . The spaces are therefore in particular vector spaces. Define ${\ displaystyle T_ {s} ^ {r} (E)}$ ${\ displaystyle E}$ ${\ displaystyle M}$ ${\ displaystyle \ pi \ colon TM \ to M}$ ${\ displaystyle T_ {p} M = \ pi ^ {- 1} (p)}$ ${\ displaystyle p \ in M}$ ${\ displaystyle T_ {p} M}$

{\ displaystyle T_ {s} ^ {r} (TM) = \ coprod _ {p \ in M} T_ {s} ^ {r} (T_ {p} M) = \ bigcup _ {p \ in M} T_ {s} ^ {r} (T_ {p} M) \ times \ {p \}}

and through with . The symbol is called coproduct . In many books, the expression on the far right is omitted. For a submanifold , the tensor bundle is defined by ${\ displaystyle \ pi _ {s} ^ {r} \ colon T_ {s} ^ {r} (TM) \ to M}$ ${\ displaystyle \ pi _ {s} ^ {r} (e) = p}$ ${\ displaystyle e \ in T_ {s} ^ {r} (T_ {p} M)}$ ${\ displaystyle \ textstyle \ coprod}$ ${\ displaystyle \ times \ {p \}}$ ${\ displaystyle A \ subset M}$

{\ displaystyle T_ {s} ^ {r} (TM) | _ {TA} = \ coprod _ {a \ in A} T_ {s} ^ {r} (T_ {a} A) = \ bigcup _ {a \ in A} T_ {s} ^ {r} (T_ {a} A) \ times \ {a \}.}

The set or the mapping are called vector bundles of tensors contravariant of level r and covariant of level s. One also speaks briefly of the tensor bundle. Whether the upper or lower index denotes the contravariance or the covariance is not uniform in the literature. ${\ displaystyle T_ {s} ^ {r} (M): = T_ {s} ^ {r} (TM)}$ ${\ displaystyle \ pi _ {s} ^ {r} \ colon T_ {s} ^ {r} (TM) \ to M}$

Tensor field

Let be a differentiable manifold. A tensor field of type (r, s) is a smooth cut in the tensor bundle . A tensor field is therefore a smooth field which assigns an (r, s) -tensor to each point of the manifold. The set of tensor fields is often referred to as. ${\ displaystyle M}$ ${\ displaystyle T_ {s} ^ {r} (M)}$ ${\ displaystyle M \ to T_ {s} ^ {r} (M)}$ ${\ displaystyle \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M))}$

Differential operators

Since a vector bundle, especially a tensor bundle, bears the structure of a manifold, the tensor field can also be understood as a smooth mapping between smooth manifolds. It is therefore possible to differentiate these fields. Differential operators that operate on smooth maps between manifolds are also known as geometric differential operators. The operators listed below fulfill the conditions of a geometric differential operator.

An important example of a differential operator operating on tensor fields is the covariant derivative . On every smooth manifold there is at least one connection , on a Riemannian manifold there is even exactly one torsion-free and metric connection , the so-called Levi-Civita connection . This relationship induces exactly one relationship on the tensor bundle, which is also called the covariant derivative. Is the underlying manifold Riemannian, so you can use the covariant derivative of the divergence differential operator by ${\ displaystyle \ nabla \ colon \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M)) \ to \ Gamma ^ {\ infty} (T_ {s} ^ {r + 1} (M) )}$

{\ displaystyle \ operatorname {div} (T): = \ operatorname {Tr} (\ xi \ mapsto \ nabla _ {\ xi} T)}

with explain.

{\ displaystyle T \ in \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M))}

The Laplace operator can also be defined for tensor fields; this is then also called the generalized Laplace operator . There are different options for defining this operator. If it is based on a Riemannian manifold, it can for example be used again by means of the covariant derivative

{\ displaystyle \ Delta T: = - \ operatorname {Tr} _ {g} \ left (\ nabla (\ nabla T) \ right)}

with explain. The figure is the tensor taper with respect to the Riemannian metric .

{\ displaystyle T \ in \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M))}

{\ displaystyle \ operatorname {Tr} _ {g}}

{\ displaystyle g}

The outer derivative , which operates on the differential forms, is also a geometric differential operator.

literature

R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .
B. Wegner: Tensor analysis . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

↑ MMG Ricci, T. Levi-Civita: Méthodes de calcul différentiel absolu et leurs applications. In: Mathematische Annalen 54, 1901, ISSN 0025-5831 , pp. 125-201, online .

[1] MMG Ricci, T. Levi-Civita: Méthodes de calcul différentiel absolu et leurs applications. In: Mathematische Annalen 54, 1901, ISSN 0025-5831 , pp. 125-201, online .