Study of Einstein condition for para-holomorphic Riemannian metrics.
problem Investigating the Einstein condition for para-holomorphic Riemannian metrics in para-complex geometry.
method Using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler Norden metrics, the paper studies the Einstein condition for para-holomorphic Riemannian metrics and their associated real para-Kähler Norden metrics.
result Every semi-simple para-complex Lie group has a natural para-Kählerian Norden Einstein metric.
Develops theory of para-holomorphic algebroids with para-complex connections.
problem Defines para-holomorphic algebroids and para-complex connections.
method Invokes Lie bialgebroids and almost para-complex structures.
result Generalizes para-Kähler geometry and Poisson-Lie groups.
Develops theory of para-holomorphic algebroids on Calabi-Yau manifolds.
problem Defines para-holomorphic algebroids on Calabi-Yau manifolds.
method Invokes Lie bialgebroids and para-complex structures to define para-holomorphic algebroids.
result Generalizes para-Kähler geometry and Poisson-Lie groups.
We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…
Investigates the connection between quadrics and Christoffel duals, and zero mean curvature surfaces.
problem Understanding the relationship between quadrics and Christoffel duals, and zero mean curvature surfaces.
method Introducing para-holomorphic elliptic functions to study timelike minimal surfaces and their Christoffel duals of 1-sheeted hyperboloids.
result Curves of type change for real isothermic surfaces of mixed causal type are aligned with the real curvature line net.
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…
Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.
problem Preserving Lagrangian condition in mean curvature flow on Kim-McCann metrics.
method Expressed mean curvature flow within generalized mean curvature flow framework.
result Lagrangian condition is preserved along the flow.
We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a parti…
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
The paper studies para-Kenmotsu manifolds and their properties.
problem Characterizing and studying properties of para-Kenmotsu manifolds.
method Using tensor equations and curvature conditions to characterize and study properties of para-Kenmotsu manifolds.
result Para-Kenmotsu manifolds with certain curvature conditions are of constant negative curvature −1. We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…
Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.
problem Extending unbalanced optimal transport to Riemannian metrics.
method Dynamic and static formulations of unbalanced optimal transport on Riemannian metrics.
result Wasserstein--Ebin metric provides a new Riemannian structure on the space of Riemannian metrics.
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodes…
Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
Riemannian metrics on orbifolds are equivalent to diffeological ones.
problem Equivalence of Riemannian and diffeological orbifolds.
method Framework of Riemannian diffeology and analysis of 2-metrics.
result Riemannian metrics on orbifolds are equivalent to diffeological ones.
Characterizes Riemannian orbifolds and their coverings via metric geometry.
problem Understanding Riemannian orbifolds and their coverings.
method Characterizes Riemannian orbifolds and their coverings via metric geometry.
result The metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and the natural projection is an orbifold covering.
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…
The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.
problem Investigating non-continuous Riemannian metrics and their infinitesimal structure.
method Constructing examples of metric measure spaces with discontinuous metrics.
result Examples show failure of infinitesimal Hilbertian or quasi-Riemannian properties.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…
The paper studies metrics on Lie groups and their curvatures.
problem Understanding left-invariant pseudo-Riemannian metrics on Lie groups.
method Formulation of a procedure based on moduli space for left-invariant pseudo-Riemannian metrics.
result Left-invariant pseudo-Riemannian metrics on Lie groups of real hyperbolic spaces have constant sectional curvatures.
Survey on metrics on compact Lie groups.
problem None explicitly stated; focuses on metrics.
method Left-invariant semi-Riemannian metrics.
result Survey of existing metrics.
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
We review and simplify the slice theorem for Riemannian metrics.
problem Existence of slices for Riemannian metrics.
method Review and concise proof of the slice theorem.
result More concise proof of slice existence.
Introduces a new framework for Riemannian diffeology.
problem No specific problem stated; focuses on a new framework.
method Uses tangent functor and metric from Iglesias-Zemmour to establish weak Riemannian diffeological spaces.
result Establishes a category of weak Riemannian diffeological spaces and shows induced pseudodistance is a distance under technical conditions.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Holomorphic metrics on complex manifolds imply infinite fundamental groups.
problem Understanding fundamental groups of complex manifolds with holomorphic metrics.
method Analyzing properties of holomorphic Riemannian metrics on compact complex manifolds.
result Compact complex manifolds with holomorphic Riemannian metrics have infinite fundamental groups.
Characterizes Zoll metrics via min-max values.
problem Characterizing Zoll Riemannian metrics.
method Uses min-max values in a loop space.
result Two min-max values coincide for Zoll metrics.
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…
We improve Riemannian metrics for constrained systems control.
problem Controlling mechanical systems with configuration constraints.
method Constructing complete Riemannian metrics by modifying incomplete ones.
result A controller can be found to satisfy a design criterion.
Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.
problem Rigidity problems for low-regularity metrics.
method Proves Cheeger-Gromoll splitting theorem and flatness criterion for semi-Riemannian metrics of C1 regularity. result Obtains isometry of higher regularity than Lipschitz.
This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…
The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.
problem Exploring Riemannian structures on tangent Lie groups.
method Defining a new left-invariant Riemannian metric on the tangent Lie group using two left-invariant metrics and symplectic forms.
result Explicit formulas for the Levi-Civita connection, tensor curvature, and sectional curvature of the new metric in terms of the original metrics.
Study sub-Riemannian metrics on compact Lie groups, finding same shortest loops.
problem Exploring sub-Riemannian length spectra on compact Lie groups.
method Restricting Killing form to root spaces of compact Lie groups.
result Same shortest loops in Riemannian and sub-Riemannian cases.
Study finds equations for spheres and circles on a specific manifold.
problem Equations for spheres and circles on a manifold with circulant structures.
method Analyzes a 3D manifold with Riemannian and additional indefinite metrics.
result Equations for spheres and circles defined with respect to the associated metric.
Study Riemannian metrics on Lie groupoids and stacks.
problem Linearize proper groupoid fibrations and derive rigidity theorems.
method Use Riemannian metrics on Lie groupoids and stacks.
result Establish Morita invariance and construct stacky tubular neighborhoods.
Riemannian metric learning improves data representation across various fields.
problem Traditional distance metrics fail to capture intrinsic data geometry.
method Leverages differential geometry to model data on Riemannian manifolds.
result Demonstrates remarkable success in diverse domains.
The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.
problem Investigating the Weyl projective rigidity of sub-Riemannian metrics.
method Analytic and smooth category proofs for specific distributions with minimal order complex abnormal extremals.
result Genericity of Weyl projectively rigid sub-Riemannian metrics and distributions.
Study geodesic distances in self-adjoint operator groups with various Riemannian metrics.
problem Geodesic distances in self-adjoint operator groups with different Riemannian metrics.
method Analysis of geodesic distances in self-adjoint operator groups with left invariant Riemannian metrics induced by infinite trace.
result Extension of completeness results to a general class of Banach-Lie groups.
Characterizes self-isometries of Riemannian metrics on compact manifolds.
problem Understanding isometries of Riemannian metrics.
method Characterization of self-isometries and proof of isometry conditions.
result Two Riemannian metric spaces are isometric if and only if their manifolds are diffeomorphic.
We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold M, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the Ck-topology, k=2,...,∞, in the set of metrics of …
Study on cut locus and cut time for a specific Riemannian metric on SO(3).
problem Finding the cut locus and cut time for a specific Riemannian metric on SO(3).
method Analytical approach to solve the metric's eigenvalue problem and derive the cut locus and cut time.
result Equation for the cut time and diameter of the metric, and description of most distant points.
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Paper calculates curvatures of metrics from isotropic structures.
problem Calculating curvatures of metrics induced by isotropic structures.
method Calculating curvature tensors of induced Riemannian metrics.
result Curvatures of the metrics are determined.
A new metric is created on a special bundle.
problem Creating a metric on a specific type of bundle.
method Lifting a metric and almost symplectic form to a supermanifold.
result A super-Sasaki metric is constructed on the antitangent bundle.
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.