A simplified proof of a theorem by Hopf.
problem Proving Hopf's theorem about hypersurface Gauss maps.
method Using the Chern-Gauss-Bonnet Theorem.
result A concise proof of Hopf's theorem.
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48 results for “Chern-Gauss-Bonnet Theorem”
We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0|n-dimensional Euclidean field theories and their quantization.
We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the Pfaffian of a complex "metric" on the tangent space and then applying analytic c…
The paper connects Chern-Gauss-Bonnet and Lefschetz Duality using a novel approach.
problem Exploring the relationship between Chern-Gauss-Bonnet and Lefschetz Duality.
method Using the mapping cone for relative deRham cohomology and Thom isomorphism.
result Shows the Chern-Gauss-Bonnet Theorem as a manifestation of Lefschetz Duality.
Derives a curvature identity on 6D manifolds with applications.
problem No specific problem stated; focuses on deriving a curvature identity.
method From the Chern-Gauss-Bonnet theorem for a 6D closed manifold.
result Derives a curvature identity on 6D manifolds.
Paper proves Liouville theorem for curvature equation with boundary conditions.
problem Proving Liouville theorem for curvature equation with boundary conditions.
method Using Chern--Gauss--Bonnet formula and boundary conditions.
result Extends Wei's earlier result by removing the infinity condition.
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
The abstract extends curvature measures to pseudo-Riemannian manifolds.
problem Extending curvature measures to pseudo-Riemannian manifolds.
method Constructing a family of generalized curvature measures.
result Generalized curvature measures behave naturally under isometric immersions.
The paper derives curvature identities for 5D and 6D Einstein manifolds.
problem Deriving curvature identities for specific dimensions of Einstein manifolds.
method Using Patterson's curvature identities and the Chern-Gauss-Bonnet Theorem, the paper provides explicit formulae for 5D and 6D Einstein manifolds.
result The curvature identities for 5D and 6D Einstein manifolds are confirmed to be consistent with previous work.
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
Extends Chern-Gauss-Bonnet formula to non-compact 4D manifolds with singularities.
problem Generalizing Chern-Gauss-Bonnet formula to non-compact 4D manifolds with singularities.
method Generalization to manifolds with conformally flat ends and singular points, under specific curvature conditions.
result Obtained a new Chern-Gauss-Bonnet formula with error terms expressed as isoperimetric deficits.
Discrete curvaturemath.DG
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
problem Defining curvature for discrete structures like polygons and polyhedral surfaces.
method Explains curvature notions for polygons, polyhedral surfaces, and abstract polyhedral manifolds.
result Discrete curvature theorems parallel classical theorems in differential geometry.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
Extended a formula to higher dimensions with singularities.
problem Generalizing a formula to higher dimensions with singularities.
method Extended a formula to all even dimensions n≥4 for a class of conformally flat manifolds with singular points.
result First formula in dimensions higher than two with isolated conical singularities.
We generalize some of the results of Harvey, Lawson and Latschev about transgression formulas. The focus here is on flowing forms via vertical vector fields, especially Morse-Bott-Smale vector fields. We prove a very general transgression formula including also a version covering non-compact situations. Among applicati…
We give a topological interpretation of the space of -harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the -Euler characteristic of some of these Manifolds. These results are applications of general theorems on complete …
Odd Pfaffian formsmath.DG
Defines odd Pfaffian form for odd-dimensional manifolds, proving Chern-Gauss-Bonnet formula.
problem Proving Chern-Gauss-Bonnet formula for incomplete edge singularities.
method Defining odd Pfaffian form through curvature tensor, proving formula for various metrics.
result Proves intrinsic Chern-Gauss-Bonnet formula for edge singularities and complete manifolds.
The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …
Formula derived for 4D manifolds with boundary involving renormalized volume and boundary integral.
problem Calculating the Euler characteristic of 4D manifolds with boundary.
method Derives a Chern-Gauss-Bonnet formula for a specific type of metric.
result Sum of renormalized volume and boundary integral is a conformal invariant when boundary is umbilic.
Mathai-Quillen forms are used to give an integral formula for the Lefschetz number of a smooth map of a closed manifold. Applied to the identity map, this formula reduces to the Chern-Gauss-Bonnet theorem. The formula is computed explicitly for constant curvature metrics. There is in fact a one-parameter family of inte…
Study calculates the renormalized area of catenoids in hyperbolic spaces.
problem Calculating the renormalized area of catenoids in hyperbolic spaces.
method Variational characterization and Chern--Gauss--Bonnet formulas for locally conformally flat manifolds.
result Renormalized area of catenoids varies continuously from negative infinity to twice the area of totally geodesic hypersurfaces.
We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that they are given solely in terms of curvature {and the second fundamental form and do not involve covariant derivatives thus general…
For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…
We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the -curvature. We show how all t…
Formula calculates manifold Euler characteristic from curvature.
problem Computing Euler characteristic from curvature invariants.
method Used hypoelliptic sub-Laplacian on forms.
result Euler characteristic can be computed from curvature invariants.
This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a…
We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in t…
This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation …
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…
This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invar…
This is the fifth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed a…
The paper finds Riemannian metric representatives for Stiefel-Whitney classes.
problem Finding representatives of Stiefel-Whitney classes using Riemannian metrics.
method Using Whitney's criteria and properties of Riemannian metrics, the paper constructs representatives for all Stiefel-Whitney classes.
result The representatives of Stiefel-Whitney classes are derived from the determinant of the metric and other geometric properties.
This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
The decomposition of global conformal invariants IV: A proposition on local Riemannian invariantsmath.DG
This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as…
This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.