Electronic Communications in Probability

The Intrinsic geometry of some random manifolds

Sunder Ram Krishnan, Jonathan E. Taylor, and Robert J. Adler

Full-text: Open access


We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 1, 12 pp.

Received: 16 December 2015
Accepted: 25 November 2016
First available in Project Euclid: 5 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 57N35: Embeddings and immersions 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G60: Random fields 70G45: Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) [See also 53Cxx, 53Dxx, 58Axx]

Gaussian process manifold random embedding intrinsic functional asymptotics

Creative Commons Attribution 4.0 International License.


Krishnan, Sunder Ram; Taylor, Jonathan E.; Adler, Robert J. The Intrinsic geometry of some random manifolds. Electron. Commun. Probab. 22 (2017), paper no. 1, 12 pp. doi:10.1214/16-ECP4763. https://projecteuclid.org/euclid.ecp/1483585770

Export citation


  • [1] Adler, R. J., Krishnan, S. R., Taylor, J. E., and Weinberger, S.: The reach of randomly embedded manifolds, arXiv:math.PR/1503.01733v1
  • [2] Adler, R. J. and Taylor, J. E.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York, 2007.
  • [3] Adler, R.J. and Taylor, J.E.: Topological complexity of smooth random functions. Springer Lecture Notes in Mathematics, vol 2019, 2011..
  • [4] Alesker, S. and Fu, J.H.G.: Integral Geometry and Valuations, Birkhäuser/Springer, 2014.
  • [5] Ledoux, M. and Talagrand, M.: Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1991.
  • [6] Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics. Springer, New York, 2006.