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Journal articles on the topic 'Curvature functionals'

Author: Grafiati

Published: 14 March 2023

Last updated: 31 July 2025

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1

Shcherbakov, Eugeniy, and Mikhail Shcherbakov. "ON THE GAUSSIAN CURVATURE FUNCTIONAL IN THE CLASS OF SURFACES OF POSITIVE GAUSSIAN CURVATURE." Mathematical Physics and Computer Simulation 27, no. 3 (2024): 60–66. https://doi.org/10.15688/mpcm.jvolsu.2024.3.5.

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The paper establishes the form of the Gaussian curvature functional defined on the class of infinitely differentiable horizontal surfaces of positive Gaussian curvature. With respect to admissible surfaces, it is assumed that they admit a global semi-geodetic parametrization. The paper proves that the first variation of the functional on the class of variations of admissible surfaces admitting connections between the coefficients of the first quadratic form and their geodesic lines similar to the axisymmetric case is determined by the Gaussian curvature of the varied surface. The consideration

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2

Ivochkina, N. M. "Minimization of functionals generating curvature operators." Journal of Soviet Mathematics 62, no. 3 (1992): 2741–46. http://dx.doi.org/10.1007/bf01670999.

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3

Sheng, Weimin, and Lisheng Wang. "Variational properties of quadratic curvature functionals." Science China Mathematics 62, no. 9 (2018): 1765–78. http://dx.doi.org/10.1007/s11425-017-9232-6.

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4

Sarkar, Prakash. "Quantifying the Cosmic Web using the Shapefinder diagonistic." Proceedings of the International Astronomical Union 11, S308 (2014): 250–53. http://dx.doi.org/10.1017/s1743921316009960.

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AbstractOne of the most successful method in quantifying the structures in the Cosmic Web is the Minkowski Functionals. In 3D, there are four minkowski Functionals: Area, Volume, Integrated Mean Curvature and the Integrated Gaussian Curvature. For defining the Minkowski Functionals one should define a surface. We have developed a method based on Marching cube 33 algorithm to generate a surface from a discrete data sets. Next we calculate the Minkowski Functionals and Shapefinder from the triangulated polyhedral surface. Applying this methodology to different data sets , we obtain interesting r

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5

Brozos‐Vázquez, Miguel, Sandro Caeiro‐Oliveira, and Eduardo García‐Río. "Critical metrics for all quadratic curvature functionals." Bulletin of the London Mathematical Society 53, no. 3 (2021): 680–85. http://dx.doi.org/10.1112/blms.12448.

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6

Kuwert, Ernst, Tobias Lamm, and Yuxiang Li. "Two-dimensional curvature functionals with superquadratic growth." Journal of the European Mathematical Society 17, no. 12 (2015): 3081–111. http://dx.doi.org/10.4171/jems/580.

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7

Joshi, Pushkar, and Carlo Séquin. "Energy Minimizers for Curvature-Based Surface Functionals." Computer-Aided Design and Applications 4, no. 5 (2007): 607–17. http://dx.doi.org/10.1080/16864360.2007.10738495.

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8

von der Mosel, Heiko. "Nonexistence results for extremals of curvature functionals." Archiv der Mathematik 69, no. 5 (1997): 427–34. http://dx.doi.org/10.1007/s000130050141.

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9

Biondi, Biondo. "Velocity estimation by image-focusing analysis." GEOPHYSICS 75, no. 6 (2010): U49—U60. http://dx.doi.org/10.1190/1.3506505.

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Migration velocity can be estimated from seismic data by analyzing, focusing, and defocusing of residual-migrated images. The accuracy of these velocity estimates is limited by the inherent ambiguity between velocity and reflector curvature. However, velocity resolution improves when reflectors with different curvatures are present. Image focusing is measured by evaluating coherency across structural dips, in addition to coherency across aperture/azimuth angles. The inherent ambiguity between velocity and reflector curvature is directly tackled by introducing a curvature correction into the co

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10

Bernardini, Marco. "New rigidity results for critical metrics of some quadratic curvature functionals." Advances in Geometry 25, no. 1 (2025): 1–12. https://doi.org/10.1515/advgeom-2024-0032.

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Abstract We prove a new rigidity result for metrics defined on closed smooth n-manifolds that are critical for the quadratic functional 𝔉 t , which depends on the Ricci curvature Ric and the scalar curvature R, and that satisfy a pinching condition of the form Sec > ε R, where ε is a function of t and n, while Sec denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying Sec > 1 48 $\begin{array}{} \displaystyle \frac{1}{48} \end{array}$ R are Einstein and, by a known result, are isometric to 𝕊4, ℝℙ4 or ℂℙ2.

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11

Blair, D. E., and D. Perrone. "A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion." Canadian Mathematical Bulletin 35, no. 4 (1992): 455–62. http://dx.doi.org/10.4153/cmb-1992-060-x.

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AbstractChern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition.

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12

Pulemotov, Artem. "Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces." Journal of Geometric Analysis 30, no. 1 (2019): 987–1010. http://dx.doi.org/10.1007/s12220-019-00175-6.

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13

Fierro, F., R. Goglione, and M. Paolini. "Finite element minimization of curvature functionals with anisotropy." Calcolo 31, no. 3-4 (1994): 191–210. http://dx.doi.org/10.1007/bf02575878.

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14

Sheng, Weimin, and Lisheng Wang. "Bach-flat critical metrics for quadratic curvature functionals." Annals of Global Analysis and Geometry 54, no. 3 (2018): 365–75. http://dx.doi.org/10.1007/s10455-018-9606-4.

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15

Moser, Roger. "Towards a variational theory of phase transitions involving curvature." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 4 (2012): 839–65. http://dx.doi.org/10.1017/s0308210510000995.

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An anisotropic area functional is often used as a model for the free energy of a crystal surface. For models of faceting, the anisotropy is typically such that the functional becomes non-convex, and then it may be appropriate to regularize it with an additional term involving curvature. When the weight of the curvature term tends to zero, this gives rise to a singular perturbation problem.The structure of this problem is comparable to the theory of phase transitions studied first by Modica and Mortola. Their ideas are also useful in this context, but they have to be combined with adequate geom

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16

Bereanu, Cristian, and Pedro J. Torres. "A Variational Approach for the Neumann Problem in Some FLRW Spacetimes." Advanced Nonlinear Studies 19, no. 2 (2019): 413–23. http://dx.doi.org/10.1515/ans-2018-2030.

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AbstractIn this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

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17

BRANSON, THOMAS, and A. ROD GOVER. "PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE." Communications in Contemporary Mathematics 09, no. 03 (2007): 335–58. http://dx.doi.org/10.1142/s0219199707002460.

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It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qk of [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qk give rise to conformally invariant quadratic forms Θk on

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18

Shojaee, Neda, and Morteza Mirmohammad Rezaii. "On the gradient flows on Finsler manifolds." International Journal of Mathematics 28, no. 01 (2017): 1750007. http://dx.doi.org/10.1142/s0129167x17500070.

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The purpose of this paper is to provide a general overview of a curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first make some differentiable structures on a domain of Finslerian functionals. Then by means of the global inner product and the Berger–Ebin Theorem, we make some decomposition for the tangent space of the manifold of all Finslerian metrics. Next, we study Akbar–Zadeh curvature functional as a Finslerian functional and we find the critical points of this functional in the pointwise conformal

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19

Arroyo, Josu, Óscar J. Garay, and Álvaro Pámpano. "Binormal Motion of Curves with Constant Torsion in 3-Spaces." Advances in Mathematical Physics 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/7075831.

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We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape.

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20

Iglesias, José A., and Alfred M. Bruckstein. "On the Gamma-convergence of some polygonal curvature functionals." Applicable Analysis 94, no. 5 (2014): 957–79. http://dx.doi.org/10.1080/00036811.2014.910302.

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21

Feoli, A., V. V. Nesterenko, and G. Scarpetta. "Functionals linear in curvature and statistics of helical proteins." Nuclear Physics B 705, no. 3 (2005): 577–92. http://dx.doi.org/10.1016/j.nuclphysb.2004.10.062.

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22

Verpoort, Steven. "Curvature functionals for curves in the equi-affine plane." Czechoslovak Mathematical Journal 61, no. 2 (2011): 419–35. http://dx.doi.org/10.1007/s10587-011-0064-4.

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23

Ma, Yuanqing, and Bing Wang. "Ricci curvature integrals, local functionals, and the Ricci flow." Transactions of the American Mathematical Society, Series B 10, no. 27 (2023): 944–87. http://dx.doi.org/10.1090/btran/155.

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Consider a Riemannian manifold ( M m , g ) (M^{m}, g) whose volume is the same as the standard sphere ( S m , g r o u n d ) (S^{m}, g_{round}) . If p > m 2 p\!>\!\frac {m}{2} and ∫ M { R c − ( m − 1 ) g } − p d v \int _{M}\! \left \{ Rc\!-\!(m\!-\!1)g\right \}_{-}^{p} dv is sufficiently small, we show that the normalized Ricci flow initiated from ( M m , g ) (M^{m}, g) will exist immortally and converge to the standard sphere. The choice of p p is optimal.

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24

Streets, Jeffrey D. "Quasi-local mass functionals and generalized inverse mean curvature flow." Communications in Analysis and Geometry 16, no. 3 (2008): 495–537. http://dx.doi.org/10.4310/cag.2008.v16.n3.a2.

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25

Kuwert, Ernst, Andrea Mondino, and Johannes Schygulla. "Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds." Mathematische Annalen 359, no. 1-2 (2014): 379–425. http://dx.doi.org/10.1007/s00208-013-1005-3.

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26

Mickel, Walter, Gerd E. Schröder-Turk, and Klaus Mecke. "Tensorial Minkowski functionals of triply periodic minimal surfaces." Interface Focus 2, no. 5 (2012): 623–33. http://dx.doi.org/10.1098/rsfs.2012.0007.

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A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. A systematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cub

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27

Bordag, M., J. Lindig, V. M. Mostepanenko, and Yu V. Pavlov. "Vacuum Stress-Energy Tensor of Nonconformal Scalar Field in Quasi-Euclidean Gravitational Background." International Journal of Modern Physics D 06, no. 04 (1997): 449–63. http://dx.doi.org/10.1142/s0218271897000261.

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The vacuum expectation value of the stress–energy tensor of a quantized scalar field with arbitrary curvature coupling in quasi-Euclidean background is calculated. The early time approximation for nonconformal fields is introduced. This approximation makes it possible to represent the matrix elements of the stress–energy tensor as explicit functionals of the scale factor. In the case of scale factors depending on time by the degree law the energy density is calculated explicitly. It is shown that the new contributions due to nonconformal curvature coupling significantly dominate the previously

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28

ARROYO, JOSU, ÓSCAR J. GARAY, and JOSE MENCÍA. "QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE." Bulletin of the Australian Mathematical Society 81, no. 3 (2010): 496–506. http://dx.doi.org/10.1017/s0004972709001142.

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AbstractThe classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the

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29

Mondino, Andrea, and Johannes Schygulla. "Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 31, no. 4 (2014): 707–24. http://dx.doi.org/10.1016/j.anihpc.2013.07.002.

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30

Anderson, Michael T. "Extrema of curvature functionals on the space of metrics on 3-manifolds." Calculus of Variations and Partial Differential Equations 5, no. 3 (1997): 199–269. http://dx.doi.org/10.1007/s005260050066.

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31

Olbermann, Heiner. "On a $\Gamma$-Limit of Willmore Functionals with Additional Curvature Penalization Term." SIAM Journal on Mathematical Analysis 51, no. 3 (2019): 2599–632. http://dx.doi.org/10.1137/18m1203596.

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32

Euh, Yunhee, JeongHyeong Park, and Kouei Sekigawa. "Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds." Differential Geometry and its Applications 29, no. 5 (2011): 642–46. http://dx.doi.org/10.1016/j.difgeo.2011.07.001.

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33

Jost, Jürgen. "Convex functionals and generalized harmonic maps into spaces of non positive curvature." Commentarii Mathematici Helvetici 70, no. 1 (1995): 659–73. http://dx.doi.org/10.1007/bf02566027.

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34

Li, Junfang. "Evolution of Eigenvalues along Rescaled Ricci Flow." Canadian Mathematical Bulletin 56, no. 1 (2013): 127–35. http://dx.doi.org/10.4153/cmb-2011-162-6.

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AbstractIn this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators -4Δ+kR is monotonic along the normalized Ricci flow for all k≥1 provided the initial manifold has nonpositive total scalar curvature.

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35

Rovenski, Vladimir. "Willmore-type variational problem for foliated hypersurfaces." Electronic Research Archive 32, no. 6 (2024): 4025–42. http://dx.doi.org/10.3934/era.2024181.

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<abstract><p>After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and

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36

Rovenski, Vladimir. "Integral Formulas for Almost Product Manifolds and Foliations." Mathematics 10, no. 19 (2022): 3645. http://dx.doi.org/10.3390/math10193645.

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Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals d

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37

Gurban, Daniela, Petru Jebelean, and Călin Şerban. "Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space." Advanced Nonlinear Studies 17, no. 4 (2017): 769–80. http://dx.doi.org/10.1515/ans-2016-6025.

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AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

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38

Ma, Wen-Xiu, Huiqun Zhang, and Jinghan Meng. "A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations." East Asian Journal on Applied Mathematics 3, no. 3 (2013): 171–89. http://dx.doi.org/10.4208/eajam.250613.260713a.

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AbstractA non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

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39

Anderson, Michael T. "Extrema of curvature functionals on the space of metrics on 3-manifolds, II." Calculus of Variations and Partial Differential Equations 12, no. 1 (2001): 1–58. http://dx.doi.org/10.1007/s005260000043.

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40

Gover, A. Rod, and Andrew Waldron. "Renormalized volumes with boundary." Communications in Contemporary Mathematics 21, no. 02 (2019): 1850030. http://dx.doi.org/10.1142/s021919971850030x.

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We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ([Formula: see text]-curvature, transgression)-type pairs for hypersurfaces wit

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41

Cuierrier, Etienne, Pierre-Olivier Roy, Rodrigo Wang, and Matthias Ernzerhof. "The fourth-order expansion of the exchange hole and neural networks to construct exchange–correlation functionals." Journal of Chemical Physics 157, no. 17 (2022): 171103. http://dx.doi.org/10.1063/5.0122761.

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The curvature Q σ of spherically averaged exchange (X) holes ρX, σ(r, u) is one of the crucial variables for the construction of approximations to the exchange–correlation energy of Kohn–Sham theory, the most prominent example being the Becke–Roussel model [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)]. Here, we consider the next higher nonzero derivative of the spherically averaged X hole, the fourth-order term T σ. This variable contains information about the nonlocality of the X hole and we employ it to approximate hybrid functionals, eliminating the sometimes demanding calcu

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42

Brito-Loeza, Carlos, and Ke Chen. "Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing." International Journal of Computer Mathematics 90, no. 1 (2013): 92–108. http://dx.doi.org/10.1080/00207160.2012.720370.

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43

Ma, Bingqing, Guangyue Huang, Xingxiao Li, and Yu Chen. "Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds." Nonlinear Analysis 175 (October 2018): 237–48. http://dx.doi.org/10.1016/j.na.2018.05.017.

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44

Pogosyan, Dmitri, Sandrine Codis, and Christophe Pichon. "Non Gaussian Minkowski functionals and extrema counts for CMB maps." Proceedings of the International Astronomical Union 11, S308 (2014): 61–66. http://dx.doi.org/10.1017/s1743921316009637.

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AbstractIn the conference presentation we have reviewed the theory of non-Gaussian geometrical measures for 3D Cosmic Web of the matter distribution in the Universe and 2D sky data, such as Cosmic Microwave Background (CMB) maps that was developed in a series of our papers. The theory leverages symmetry of isotropic statistics such as Minkowski functionals and extrema counts to develop post Gaussian expansion of the statistics in orthogonal polynomials of invariant descriptors of the field, its first and second derivatives. The application of the approach to 2D fields defined on a spherical sk

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45

Zafar, Sana, and Maria Hussain. "Fair curve designing by Said-Ball curve." PLOS One 20, no. 7 (2025): e0324553. https://doi.org/10.1371/journal.pone.0324553.

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Fair curves are visually alluring curves and are free of unnecessary design features. A new curve designing method is introduced using the tangential continuous rational cubic Said-Ball curve. Fair curves are achieved by controlling its length and variation in curvature. It has enough degrees of freedom (control points and free parameters). A family of curves can be obtained for different choices for the values of free parameters. The control points are fixed by employing the G1 continuity conditions at the end points of the RCSBC. The optimal values of the remaining free parameters that are t

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46

KAWAI, EI-ICHIRO. "A FURTHER COMMENT ON THE HAMILTON FORMALISM FOR NONLINEAR INTEGRABLE MODELS." Modern Physics Letters A 08, no. 31 (1993): 2919–26. http://dx.doi.org/10.1142/s0217732393003330.

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An attractive operator equation, explicated in the preceding work,1 is intensively investigated with careful attention paid to its characteristics originated by alternate action of dual Hamiltonian operators. In this context, it is argued that its amenable modification can be thought of as a sort of null curvature equation. On the basis of such intriguing view, an application of the fruitful gauge-theoretic concept is tried, from which a novel formula for obtaining systematically the conserved Hamiltonian functionals is derived as a by-product.

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47

Mourier, Pierre, and Asta Heinesen. "Splitting the spacetime: a systematic analysis of foliation dependence in cosmic averaging." Journal of Cosmology and Astroparticle Physics 2024, no. 04 (2024): 067. http://dx.doi.org/10.1088/1475-7516/2024/04/067.

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Abstract It is a fundamental unsolved question in general relativity how to unambiguously characterize the effective collective dynamics of an ensemble of fluid elements sourcing the local geometry, in the absence of exact symmetries. In a cosmological context this is sometimes referred to as the averaging problem. At the heart of this problem in relativity is the non-uniqueness of the choice of foliation within which the statistical properties of the local spacetime are quantified, which can lead to ambiguity in the formulated average theory. This has led to debate in the literature on how to

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48

Fröhlich, Steffen. "On two-dimensional immersions that are stable for parametric functionals of constant mean curvature type." Differential Geometry and its Applications 23, no. 3 (2005): 235–56. http://dx.doi.org/10.1016/j.difgeo.2005.05.005.

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49

Kang, J. H., and A. W. Leissa. "Three-Dimensional Field Equations of Motion, and Energy Functionals for Thick Shells of Revolution With Arbitrary Curvature and Variable Thickness." Journal of Applied Mechanics 68, no. 6 (2001): 953–54. http://dx.doi.org/10.1115/1.1406961.

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Equations of motion and energy functionals are derived for a three-dimensional coordinate system especially useful for analyzing the static and dynamic behavior of arbitrarily thick shells of revolution having variable thickness. The field equations are utilized to express them in terms of displacement components.

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50

Blair, David E. "A Survey of Riemannian Contact Geometry." Complex Manifolds 6, no. 1 (2019): 31–64. http://dx.doi.org/10.1515/coma-2019-0002.

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AbstractThis survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on c

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