Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Orthogonal geodesic and minimal distributions


Author: Irl Bivens
Journal: Trans. Amer. Math. Soc. 275 (1983), 397-408
MSC: Primary 53C12; Secondary 57R30
DOI: https://doi.org/10.1090/S0002-9947-1983-0678359-4
MathSciNet review: 678359
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{F}$ be a smooth distribution on a Riemannian manifold $ M$ with $ \mathfrak{H}$ the orthogonal distribution. We say that $ \mathfrak{F}$ is geodesic provided $ \mathfrak{F}$ is integrable with leaves which are totally geodesic submanifolds of $ M$. The notion of minimality of a submanifold of $ M$ may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to $ \mathfrak{H}$ then we say $ \mathfrak{H}$ is minimal. Suppose that $ \mathfrak{F}$ and $ \mathfrak{H}$ are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of $ \mathfrak{F}$ is also a submanifold of Euclidean space with mean curvature normal vector field $ \eta $. We show that the integral of $ \vert\eta {\vert^2}$ over $ M$ is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold.

We study the relationships between the geometry of $ M$ and the integrability of $ \mathfrak{H}$. For example, if $ \mathfrak{F}$ and $ \mathfrak{H}$ are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then $ \mathfrak{H}$ is integrable iff $ \mathfrak{F}$ and $ \mathfrak{H}$ are parallel distributions. Similarly if $ {M^n}$ has constant negative sectional curvature and dim $ \mathfrak{H} = 2 < n$ then $ \mathfrak{H}$ is not integrable. If $ \mathfrak{F}$ is geodesic and $ \mathfrak{H}$ is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of $ \mathfrak{H}$ are flat submanifolds of $ M$ with parallel second fundamental form.


References [Enhancements On Off] (What's this?)

  • [1] I. Bivens, Codazzi tensors and reducible submanifolds, Trans. Amer. Math. Soc. 268 (1981), 231-246. MR 628456 (82k:53072)
  • [2] R. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), 143-158. MR 558593 (81h:57011)
  • [3] -, Foliated manifolds with flat basic connection, J. Differential Geom. 16 (1981), 401-406. MR 654633 (83g:57015)
  • [4] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-737. MR 0205184 (34:5018)
  • [5] H. Huck, R. Roitzsch, U. Simon, W. Vortisch, R. Walden, B. Wegner and W. Wendland, Beweismethoden der Differential Geometrie im Grossen, Springer-Verlag, Berlin and Heidelberg, 1973. MR 0370439 (51:6666)
  • [6] D. Johnson, Kähler submersions and holomorphic connections, J. Differential Geom. 15 (1980), 71-79. MR 602440 (82f:53065)
  • [7] D. Johnson and L. Whitt, Totally geodesic foliations, J. Differential Geom. 15 (1980), 225-235. MR 614368 (83h:57037)
  • [8] B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. MR 0200865 (34:751)
  • [9] R. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977), 525-533. MR 0482597 (58:2657)
  • [10] B. Wegner, Codazzi-tensoren und kennzeichnungen sphärischer immersionen, J. Differential Geom. 9 (1974), 61-70. MR 0362158 (50:14600)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C12, 57R30

Retrieve articles in all journals with MSC: 53C12, 57R30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678359-4
Keywords: Geodesic distribution, minimal distribution, parallel distribution, foliation, minimal submanifold, Codazzi tensor
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society