A generalization of parallelism in Riemannian geometry, the case

Author:
Alan B. Poritz

Journal:
Trans. Amer. Math. Soc. **152** (1970), 461-494

MSC:
Primary 53.72

DOI:
https://doi.org/10.1090/S0002-9947-1970-0268813-2

MathSciNet review:
0268813

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The concept of parallelism along a curve in a Riemannian manifold is generalized to parallelism along higher dimensional immersed submanifolds in such a way that the minimal immersions are self parallel and hence correspond to geodesics. Let be a (not necessarily isometric) immersion of Riemannian manifolds. Let be a tangent bundle isometry along , that is, covers and maps fibers isometrically. By mimicing the construction used for isometric immersions, it is possible to define the mean curvature vector field of is said to be parallel along if this vector field vanishes identically. In particular, minimal immersions have parallel tangent maps. For curves, it is shown that the present definition reduces to the definition of Levi-Civita. The major effort is directed toward generalizations, in the real analytic case, of the two basic theorems for parallelism. On the one hand, the existence and uniqueness theorem for a geodesic in terms of data at a point extends to the well-known existence and uniqueness of a minimal immersion in terms of data along a codimension one submanifold. On the other hand, the existence and uniqueness theorem for a parallel unit vector field along a curve in terms of data at a point extends to a local existence and uniqueness theorem for a parallel tangent bundle isometry in terms of mixed initial and partial data. Since both extensions depend on the Cartan-Kahler Theorem, a procedure is developed to handle both proofs in a uniform manner using fiber bundle techniques.

**[1]**W. Ambrose,*Higher order Grassmann bundles*, Topology**3**(1965), no. suppl, suppl. 2, 199–238. MR**171287**, https://doi.org/10.1016/0040-9383(65)90076-5**[2]**Richard L. Bishop and Richard J. Crittenden,*Geometry of manifolds*, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR**0169148****[3]**N. Bourbaki,*Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7)*, Actualités Scientifiques et Industrielles, No. 1333, Hermann, Paris, 1967 (French). MR**0219078****[4]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[5]**H. H. Johnson,*Systems of differential forms including Kuranishi's theory of total prolongations*, Dept. of Math., Univ. of California, Berkeley, Technical Report #3, U. S. Army Office of Ordinance Research.**[6]**J.-L. Koszul,*Lectures on fibre bundles and differential geometry*, Notes by S. Ramanan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 20, Tata Institute of Fundamental Research, Bombay, 1965. MR**0268801****[7]**Alan B. Poritz,*A generalization of parallelism in riemannian geometry; the 𝐶^{∞} case*, J. Differential Geometry**4**(1970), 187–192. MR**264566****[8]**Alan B. Poritz,*A global existence and uniqueness theorem for generalized parallelism.*, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 293–298. MR**0264567****[9]**James Simons,*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**233295**, https://doi.org/10.2307/1970556

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53.72

Retrieve articles in all journals with MSC: 53.72

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0268813-2

Keywords:
Parallelism,
auto-parallel,
least-area variational problem,
geodesic,
minimal immersion,
connection,
Levi-Civita connection,
vector bundle,
Euclidean vector bundle map,
tangent bundle isometry,
Grassmann manifold,
oriented Grassmann manifold,
canonical vector bundle,
bundle of adapted frames,
isometric immersion,
second fundamental form,
mean curvature vector field,
parallel vector field,
parallel tangent bundle isometry,
differential ideal,
vector bundle valued form,
connection form,
equivariant form,
regular integral plane,
polar space,
Cartan-Kahler Theorem,
initial data problem,
bundle of partial data

Article copyright:
© Copyright 1970
American Mathematical Society