The study constructs immersions with controlled curvatures between manifolds and identifies obstacles.
problem Creating immersions with manageable curvatures between Riemannian manifolds.
method Develops techniques to construct immersions with controlled curvatures and identifies obstructions.
result Identifies conditions under which immersions with specific curvature constraints are possible.
An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…
The paper classifies immersed 3-manifolds bounded by a surface in 3D space.
problem Classifying immersed 3-manifolds bounded by a surface in 3D space.
method Determining equivalence classes of immersed 3-manifolds.
result All equivalence classes of immersed 3-manifolds bounded by an arbitrary immersed surface in R3 are determined. Study immersions of Sasakian manifolds into Sasakian space forms.
problem Characterize Sasakian manifolds that can be immersed into Sasakian space forms.
method Proved Calabi's rigidity results, characterised homogeneous Sasakian manifolds.
result Characterized homogeneous Sasakian manifolds that can be embedded into the standard sphere.
Algorithm decides if odd-dimensional maps can be immersed.
problem Deciding if maps can be immersed homotopic to a given map.
method Algorithm for odd codimension manifolds.
result Algorithm decides immersibility of maps in odd codimension.
Study immersions of punctured 4-manifolds for quantum automata applications.
problem Existence of immersions between specific 4-manifolds.
method Analyzing immersions of punctured 4-manifolds to establish a partial order.
result Established a partial order on closed 4-manifolds via immersions.
Algorithmic study of manifold immersibility and embeddability.
problem Decidability of manifold immersibility and embeddability.
method Algorithmic analysis of immersion and embeddability questions for m-dimensional CAT-manifolds in Rn. result Undecidability of smooth embeddability when n−m is even and 11m≥10n+1. New examples show immersions not homologous to embeddings.
problem Tackles the question of immersions vs. embeddings in manifolds.
method Provides examples showing immersions not homologous to embeddings.
result Double points represent non-trivial homology classes.
The paper classifies Sasakian immersions into odd spheres and finds new examples.
problem Classifying Sasakian immersions into odd-dimensional spheres.
method Analyzing compact Sasakian manifolds and using classification theorems.
result Infinite families of compact Sasakian η-Einstein manifolds cannot be immersed into odd spheres. Paper explores conformal immersions of Kaehler manifolds into Euclidean space.
problem Understanding conformal immersions of Kaehler manifolds.
method Used techniques from S. Chion and M. Dajczer for hyperbolic space immersions.
result Proved properties of conformal immersions into Euclidean space.
The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we obtain conditions for realizing an n-dimensional statistical manifold in an (n+1)-dim…
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces im…
Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.
problem Isometric immersions of S² into 3D Riemannian manifolds with non-negative Gauss curvature.
method Utilizes the framework of J-holomorphic curves developed by Labourie.
result Exhibits a sufficient condition for the existence of global C¹¹ isometric immersions.
Estimates warping functions for isometric immersions on specific Riemannian manifolds.
problem Estimating warping functions for isometric immersions.
method Using results from \cite{P1}, estimates are derived by changing target manifolds to constant space forms and Hermitian symmetric spaces.
result Obtained estimates and equality cases for warping functions.
We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of …
Investigates immersions in Sn using complex spinors.
problem Generalizing immersions of Riemann surfaces to Spin-manifolds.
method Uses complex spinors and the Dirac equation.
result Investigates submanifolds of SpinC-manifolds of constant curvature.
Study minimal Kähler submanifolds in product of space forms.
problem Existence and properties of minimal isometric immersions of Kähler manifolds into product spaces.
method Analyse obstruction conditions and prove existence results for minimal immersions into Sm−1imesR and Hm−1imesR. result Only minimal isometric immersions of Riemannian surfaces into Sm−1imesR and Hm−1imesR exist. Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answe…
Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified. Study CR immersions into Kähler manifolds, proving new theorems.
problem Understanding CR immersions and umbilical points in complex geometry.
method Analysis of second fundamental forms and eigenvalues of the Kohn Laplacian.
result Extension of Webster's theorem and new gap theorem.
Study finds all conformal minimal immersions of 2-spheres in a complex Grassmann manifold with parallel second fundamental form.
problem Classifying conformal minimal immersions with parallel second fundamental form.
method Analyzing immersions in complex Grassmann manifold G(2,N;C). result Determined all conformal minimal immersions of 2-spheres with parallel second fundamental form.
Researchers find a list of non-isometric toric para-Kaehler-Einstein manifolds.
problem Finding a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form.
method Analytical approach to find mutually non-isometric toric para-Kaehler-Einstein manifolds.
result A list of mutually non-isometric toric para-Kaehler-Einstein manifolds analytically immersed in a finite-dimensional para-Kaehler space form.
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
The paper proves existence and uniqueness of slant immersions in complex space forms.
problem Existence and uniqueness of slant immersions in complex space forms.
method Established existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
result Existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space Rn in terms of an integral representation of Weierstrass type. Restricting to…
Study on constant curvature immersions of surfaces into flag manifolds.
problem Investigate constant curvature immersions of Riemann surfaces into flag manifolds.
method Investigate pseudoholomorphic maps and invariant metrics on flag manifolds.
result Unitarily equivalent primitive immersions of the two-sphere into full flag manifolds have constant curvature under all invariant metrics.
Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of …
Approximates compact and non-compact Sasakian manifolds in spheres.
problem Approximating Sasakian structures in spheres.
method CR immersions in standard spheres.
result Compact and non-compact Sasakian manifolds can be approximated.
In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds.
The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.
problem Investigating the geometry of gradient almost Yamabe solitons in warped product manifolds.
method Presenting geometric rigidity results, investigating existence conditions, and classifying specific solitons.
result Classification of rotational gradient almost Yamabe solitons in RimesfRn. The paper proves conditions for 4-manifolds to be parallelizable and non-existence results.
problem Conditions for 4-manifolds to be parallelizable and non-existence results.
method Homotopy theory, particularly obstruction theory.
result Compact, oriented 4-manifolds with vanishing Euler characteristic and signature admit coassociative-free immersions.
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.
This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersio…
Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an an…
Recover manifold homology from a sample in Euclidean space.
problem Recovering the homology of an immersed manifold from a sample.
method Estimate tangent bundle, apply measure-based filtrations for persistent homology, use normal reach.
result Construction consistent and stable, does not require manifold dimension.
An isometric immersion of a Riemannian manifold M into a Riemannian manifold N gives rise in a natural way to variety of immersions into the tangent bundle TN with a non-degenerate g− natural metric G. In the paper we introduce and study an immersion into TN defined by the immersion f:M⟶N i…
Local Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds.
problem Characterizing local immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds.
method Analyzing local Sasakian immersions of Sasaki-Ricci solitons into fiber products of homogeneous Sasakian manifolds.
result Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds. (CMC) 1-immersions in hyperbolic 3-manifolds often develop singularities.
problem Existence and uniqueness of (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds.
method Analysis of blow-up phenomena and orthogonality conditions.
result Existence and uniqueness of (CMC) 1-immersions for surfaces of any genus.
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: an…
The paper shows that immersions of hypersurfaces can fail to be manifolds and converges under certain conditions.
problem Non-compactness of W2,d immersions of d-dimensional hypersurfaces. method Construction of a family of immersions and analysis of their limits.
result A family of immersions whose limit fails to be an immersion of a manifold.
Study shows how compact shapes can be rigidly mapped into complete manifolds.
problem Rigidity of isometric immersions in complete manifolds.
method Local quantitative rigidity estimates, reduced to Euclidean setting.
result Subsequence of immersions converges to an isometric immersion.
The paper finds inequalities for eigenvalues of operators on immersed manifolds.
problem Finding inequalities for eigenvalues of operators on immersed manifolds.
method Computing inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space.
result Universal inequalities for eigenvalues of operators are computed.
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew symmetric second fundamental form, if and onl…
Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.
problem Characterizing Kähler-Ricci solitons that can be immersed into complex space forms.
method Proving that a Kähler-Ricci soliton's metric is Einstein if it can be immersed into a complex space form.
result The Kähler-Ricci soliton's metric is an Einstein metric if it can be immersed into a complex space form.