View source for Parallel field

''covariantly-constant field''

A tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713501.png" /> (cf. [[Tensor analysis|Tensor analysis]]) on a [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713502.png" /> with a [[Linear connection|linear connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713503.png" /> that is invariant under [[Parallel displacement(2)|parallel displacement]] along curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713504.png" />. This means that for any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713505.png" /> the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713506.png" /> (the value of the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713507.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713508.png" />) under the parallel displacement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p0713509.png" /> along any smooth curve joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135011.png" /> transforms to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135012.png" />.

A tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135013.png" /> is parallel if and only if its [[Covariant derivative|covariant derivative]] in the direction of any vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135014.png" /> vanishes identically: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135015.png" /> or, in other words, if the [[Covariant differential|covariant differential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135017.png" /> vanishes.

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135018.png" /> of parallel fields forms a subalgebra of the algebra of all tensor fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135019.png" /> that is invariant under contraction of tensor fields and permutation of their indices. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135020.png" /> is naturally isomorphic to the algebra of tensors at a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135022.png" /> that are invariant under the homogeneous [[Holonomy group|holonomy group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135024.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135025.png" />. For a connection with full holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135027.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135028.png" /> is generated by the Kronecker symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135029.png" />; for a [[Riemannian connection|Riemannian connection]] with holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135030.png" /> — by the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135031.png" /> and its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135032.png" />, and for a Riemannian connection with holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135033.png" /> — by the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135035.png" /> and the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135036.png" />-form. There are also descriptions of the generators of the algebra of parallel differential forms on an arbitrary space with a torsion-free linear connection with an arbitrary irreducible holonomy group [[#References|[5]]].

Of special interest are the parallel fields of differential forms on a Riemannian manifold with a [[Levi-Civita connection|Levi-Civita connection]]. With every such form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135037.png" /> there is associated (by means of the tensor product with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135038.png" /> and contractions) a number of linear operators in the space of differential forms that commute with the Laplace–Beltrami operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135039.png" />, for example, the operator of interior and exterior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135040.png" /> or the operator of orthogonal projection onto subspaces of the space of differential forms that are invariant relative to the holonomy group. The study of these operators makes it possible to obtain estimates for the dimensions of the spaces of harmonic forms of various degrees, that is, (in the compact case) for the Betti numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135041.png" /> (see [[#References|[4]]]). The most substantial theory (see [[Hodge theorem|Hodge theorem]]) has been developed for Kähler and for quaternion Kähler manifolds (cf. [[Kähler manifold|Kähler manifold]]), in which there always is a parallel field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135042.png" />-forms, respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135043.png" />-forms. Any parallel differential form in a Riemannian space is harmonic. In a compact symmetric Riemannian space the converse is true: Any [[Harmonic form|harmonic form]] is parallel. Therefore, the real cohomology ring of a compact symmetric space is isomorphic to the ring of parallel differential forms.

A tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135044.png" /> is a parallel field, relative to some linear connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135045.png" />, if and only if it is infinitesimally homogeneous, that is, if at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135047.png" /> there is a frame relative to which the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135048.png" /> has fixed components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135049.png" /> independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135050.png" />. In this case the set of frames relative to which the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135052.png" />, have the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135053.png" /> form a [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135054.png" />-structure]], that is, the principal subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135055.png" /> of the bundle of frames with the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135057.png" /> is the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135058.png" /> under the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135059.png" /> on the tensor space. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135060.png" /> is parallel relative to any connection in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135061.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135062.png" />. In particular, any section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135063.png" /> (if it exists) gives a connection with zero curvature relative to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135064.png" /> is a parallel field.

More complicated is the problem of the existence of torsion-free connections relative to which a given infinitesimally-homogeneous field is parallel. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135065.png" /> has a pseudo-Riemannian metric, then such a connection (a Levi-Civita connection) always exists, and is unique. It turns out that this case is exceptional: If for any infinitesimally-homogeneous tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135066.png" /> of certain type there exists a unique torsion-free connection relative to which it is parallel, then the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135067.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135068.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135069.png" /> is pseudo-orthogonal, and consequently a pseudo-Riemannian metric is canonically associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135070.png" /> [[#References|[7]]]. For a broad class of infinitesimally-homogeneous tensor fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135071.png" /> the presence of a torsion-free connection relative to which the field is parallel implies the integrability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135072.png" />, that is, the existence of a local coordinate system in which the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135073.png" /> are constant. This is so, for example, for an almost-complex structure, for an almost-symplectic structure and for any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135074.png" /> for which the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135075.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135076.png" /> is irreducible and does not belong to the known list of irreducible holonomy groups of spaces with a torsion-free linear connection.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1–2''' , Interscience (1963–1969) {{MR|1393941}} {{MR|1393940}} {{MR|0238225}} {{MR|1533559}} {{MR|0152974}} {{ZBL|0526.53001}} {{ZBL|0508.53002}} {{ZBL|0175.48504}} {{ZBL|0119.37502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French) {{MR|0413000}} {{ZBL|0337.53031}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.-S. Chern, "On a generalization of Kähler geometry" R.H. Fox (ed.) D.C. Spencer (ed.) A.W. Tucker (ed.) , ''Algebraic geometry and topology (Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 103–121 {{MR|0087172}} {{ZBL|0078.14103}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Berger, "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes" ''Bull. Soc. Math. France'' , '''83''' (1955) pp. 279–330 {{MR|0079806}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) {{MR|0355886}} {{ZBL|0246.53031}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On a fundamental theorem of Weyl–Cartan on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071350/p07135077.png" />-structures" ''J. Math. Soc. Japan'' , '''17''' (1965) pp. 84–101 {{MR|192438}} {{ZBL|}} </TD></TR></table>