Low Dimensional Flat Manifolds with Some Elasses of Finsler Metric by Sedigheh Alavi Endrajemi, Mehdi Rafie-Rad

“…A classical result affirms that a Riemannian manifold is flat if and only if its Riemann curvature vanishes (equivalently, the sectional curvature; This is usually taken as the definition of a flat Riemannian manifold in the contexts. …”
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Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces by Sun, Ao

“…Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. …”
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Fluids, geometry, and the onset of Navier–Stokes turbulence in three space dimensions by Chen, G, Slemrod, M, Wang, D

“…The Nash-Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving C1 manifold. …”

Generalized Semi-Symmetric Non-Metric Connections of Non-Integrable Distributions by Tong Wu, Yong Wang

“…In this work, the cases of non-integrable distributions in a Riemannian manifold with the first generalized semi-symmetric non-metric connection and the second generalized semi-symmetric non-metric connection are discussed. …”
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Ollivier-Ricci curvature convergence in random geometric graphs by Pim van der Hoorn, William J. Cunningham, Gabor Lippner, Carlo Trugenberger, Dmitri Krioukov

“…Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. …”
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Quantitative Topology of Loop Space by Elliott, Robin

“…In this thesis we investigate how the size of a cycle in the based loop space of a simply connected Riemannian manifold controls its topology. Analogous to Gromov’s notion of distortion of higher homotopy groups of underlying Riemannian manifold, we define notions of distortion for (co)homology classes in the loop space with real coefficients, and study the asymptotics of these distortions. …”
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Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature by Ikoma, N, Malchiodi, A, Mondino, A

“…Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. …”

A study of horizontally weakly conformal maps and their distributions by Mehran Aminian

“…Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. …”
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A Note on Incompressible Vector Fields by Nasser Bin Turki

“…We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. …”
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When Will a Sequence of Points in a Riemannian Submanifold Converge? by Tuyen Trung Truong

“…Let X be a Riemannian manifold and <inline-formula><math display="inline"><semantics><msub><mi>x</mi><mi>n</mi></msub></semantics></math></inline-formula> a sequence of points in X. …”
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Reducing the Dimensionality of SPD Matrices with Neural Networks in BCI by Zhen Peng, Hongyi Li, Di Zhao, Chengwei Pan

“…In brain–computer interface (BCI)-based motor imagery, the symmetric positive definite (SPD) covariance matrices of electroencephalogram (EEG) signals with discriminative information features lie on a Riemannian manifold, which is currently attracting increasing attention. …”
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On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure by Esmaeil Peyghan, Farshad Firuzi

“…On the other hand, many of mathematicians have widely considered the concept of lifted metric on the tangent bundle and tangent sphere bundle of a Riemannian manifold (M, g). The idea of constructing a lifted metric on the tangent bundle was a strong inspiration for many of mathematicians and finally, the notion of g-natural metric as the most general type of lifted metrics on tangent bundle TM of a Riemannian manifold (M, g) was introduced in 2005. …”
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On global existence for variants of Teichmueller harmonic map flow by Robertson, C

“…<p>We resolve questions of existence and singularity formation for the Teichmueller harmonic map flow, a geometric flow introduced by Rupflin and Topping that deforms surfaces immersed in a Riemannian manifold into minimal surfaces.</p> <p>We obtain global solutions of Teichmueller harmonic map flow in new settings from cylinders into nonpositively curved targets, and prove their asymptotic convergence to solutions of the Douglas–Plateau problem of finding minimal surfaces spanning given boundary curves in a Riemannian manifold. …”

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations by Shkoller, S

“…We present a very simple proof of the global existence of a $C^\infty$ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has $C^\infty$ dependence on initial data $u_0$ in the class of $H^s$ divergence-free vector fields for $s&gt;2$.…”

On Harnack estimates for positive solutions of the heat equation on a complete manifold by Bakry, D, Qian, Z

“…We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. …”

Geometric quantization and dynamical constructions on the space of Kähler metrics by Rubinstein, Yanir Akiva

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A CONSTRUCTION OF DIFFEOMORPHISM EXTENSION AND ITS APPLICATIONS by A. M. Lukatsky

“…Let M be Riemannian manifold with boundary and a diffeomorphism of . …”
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Mean Curvature Flow, Orbits, Moment Maps by Pacini, T

“…Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. …”

Nonexistence of DEC spin fill-ins by Raulot, Simon

“…In this note, we show that a closed spin Riemannian manifold does not admit a spin fill-in satisfying the dominant energy condition (DEC) if a certain generalized mean curvature function is point-wise large.…”
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Torse-forming vector fields on m -spheres by Amira Ishan, Sharief Deshmukh

“…A characterization of an $ m $-sphere $ \mathbf{S}^{m}(a) $ is obtained using a non-trivial torse-forming vector field $ \zeta $ on an $ m $-dimensional Riemannian manifold.…”
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