Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
Abstract reviews geometric theories of smooth and F-smooth systems.
problem Geometric theories of smooth and F-smooth systems.
method Reviews geometric theories of smooth and F-smooth systems.
result Discusses geometric theories of smooth and F-smooth systems.
We prove that smooth cube manifolds have normal smooth structures.
Confirming operator characterization on smooth manifolds.
problem Characterizing an operator on smooth sections of tangent bundles.
method Using algebraic axioms and H^1(M, R) = {0} assumption.
result Operator can be characterized universally for any smooth manifold.
Smooth manifolds from locally homogeneous spaces.
problem Understanding smoothness in C0-Riemannian manifolds. method Demonstrated locally homogeneous C0-Riemannian manifolds are smooth. result Locally homogeneous C0-Riemannian manifolds are smooth. Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.
problem Understanding smoothness in group actions on 3-manifolds.
method Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds. result Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds are conjugate to smooth actions. Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Smooth deformation of Moishezon manifolds preserves their Moishezon property.
problem Preserving Moishezon property under smooth deformation.
method Smooth deformation over a unit disk in C.
result Deformation limit of Moishezon manifolds is Moishezon.
The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.
problem Classifying smooth structures on product manifolds.
method Computational and classification methods for concordance and diffeomorphism.
result Diffeomorphism classification of MimesS1 for specific M and k. The paper constructs infinitely many G-smoothings of a G-manifold.
problem Constructing G-smoothings of a G-manifold. method Using controlled h-cobordisms. result Infinitely many G-smoothings of a G-manifold are constructed and are isotopic after taking a product with R. In this paper we prove several related results concerning smooth Zp or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds Xn, n≥2, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each Xn support…
We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein …
Characterizes smooth functions on manifolds with simple Reeb spaces.
problem Understanding the structure of Reeb spaces for smooth functions.
method Analyzes smooth functions on closed manifolds to determine their Reeb spaces' structure.
result Characterizes smooth functions whose Reeb spaces are finite graphs.
Entropy data replaces classical charts for smooth manifolds.
problem Establishing smooth structures on topological manifolds.
method Using entropy data to define admissible coordinate functions and reconstruct smooth atlases.
result Entropy-smooth structures are equivalent to classical smooth structures and stable under perturbations.
The paper explores conditions for homology spheres to bound acyclic smooth manifolds and symplectic fillings.
problem Conditions for integral homology 3-spheres to bound acyclic smooth 4-manifolds and their symplectic fillings.
method Structural results and analysis of smooth embeddings of lens spaces in C2. result Smooth embeddings of connected sums of lens spaces in C2 cannot be upgraded to Stein embeddings. New smooth structures found on certain 4D spaces.
problem Finding distinct smooth structures on specific 4D spaces.
method Constructing specific 4-manifolds with infinite fundamental group.
result Infinitely many non-diffeomorphic structures found.
The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.
problem Exploring smooth structures on four-manifolds with finite cyclic fundamental groups.
method Analyzes topological four-manifolds with odd intersection forms and diverse fundamental groups.
result Many four-manifolds with cyclic fundamental groups admit infinitely many distinct smooth structures.
The paper smooths out equations on 4-manifolds to show smooth moduli spaces.
problem Understanding the structure of solutions to Vafa-Witten equations on 4-manifolds.
method Constructing perturbation terms to ensure transversality and showing moduli spaces are smooth.
result For generic perturbation, the general part of the moduli space is a smooth 0-dimensional manifold.
This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold C∞(M,N) of all smooth mappings from a finite dimensional Whitney manifold germ M into a …
Let M be any n dimensional smooth manifold and PM be the space of all smooth paths, then we showed that PM is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its relation to ambient space.
Every 4-manifold can be smoothly embedded in complex projective 3-space.
problem Embedding 4-manifolds in complex projective spaces.
method Analyzing properties of 4-manifolds and complex projective spaces.
result Every closed orientable smooth 4-manifold admits a smooth embedding in $\CP^3$.
Among all C∞-algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds.
The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.
problem Understanding the smooth mapping class groups of certain 4-manifolds.
method Geometric and Teichmüller-theoretic methods.
result Proves a global Torelli theorem for generalized Enriques manifolds.
We prove a reflection principle for minimal surfaces in smooth (non necessarily analytic) three manifolds and we give an explicit application when the ambient space is just a smooth manifold.
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be C0-approximated by PL symplectic manifolds. New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.
We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Smooth manifolds can be triangulated with graphs of bounded twin-width.
problem Understanding the structure of triangulations of smooth manifolds.
method Using Whitney's triangulation method and bounding the twin-width of specific graphs.
result Compact smooth manifolds have triangulations with graphs of bounded twin-width.
We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
The study finds smooth structures on specific 4-manifolds with even fundamental groups.
problem Investigating smooth structures on 4-manifolds with even fundamental groups.
method Analyzing topological, closed, oriented, non-spin 4-manifolds with given constraints.
result Existence of either none or infinitely many distinct smooth structures on specified manifolds.
In this paper we present another notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. Afterwards we introduce complex manifolds with corners and show that if M is a compact (respectively complex) manifold with corners and K is a smooth (respectively complex) Lie gro…
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…
Variant of previous work on smooth algebraic functions with compact and non-compact preimages.
problem Constructing smooth algebraic functions with specific preimage properties.
method Explicit construction of smooth real algebraic functions with controlled preimage compactness.
result New results in singularity theory and real algebraic geometry.
Embeds 3-manifolds in symplectic 4-manifolds with constraints.
problem Embedding 3-manifolds in symplectic 4-manifolds with topological and smooth properties.
method Topological and smooth embeddings, using homology cobordism and obstructions.
result 3-manifolds can be embedded in symplectic 4-manifolds with specific conditions.
Notes on embedding criteria for smooth manifolds.
problem Conditions for embedding smooth manifolds into Euclidean space.
method Linking recent results to classical criteria and K-theory.
result Recent results connect to classical embedding criteria.
Classifies Real line bundles with Real connections on manifolds with involution.
problem Classifying Real line bundles with Real connections on manifolds with involution.
method Defines Real smooth Deligne cohomology to interpolate between equivariant sheaf cohomology and smooth imaginary-valued forms.
result Classifies Real line bundles with Real connections on manifolds with involution.
We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on t…
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…
It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R4. Similarly, a smooth 4-manifold homeomorphic to the produc…
The paper explores the structure of Reeb spaces for smooth functions on manifolds.
problem Understanding the structure of Reeb spaces for smooth functions on manifolds.
method Proving the structure of Reeb spaces and showing that any graph can be realized as a Reeb space.
result The Reeb space of a smooth function on a closed manifold with finitely many critical values has a graph structure.
Study on lens spaces bounding 4-manifolds with specific Betti numbers.
problem Which lens spaces can bound 4-manifolds with second Betti number one?
method Construction of specific 4-manifolds and analysis of lens space boundaries.
result Infinite families of lens spaces can bound 4-manifolds with second Betti number one, but not all.
Diffusion models adapt to data geometry through log-domain smoothing.
problem Understanding why diffusion models generalize well across diverse domains.
method Investigating the role of score matching and log-domain smoothing in diffusion models.
result Log-domain smoothing adapts the diffusion model to the data manifold.
In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an ap…