The standard smooth step function

Depends on

• The standard flat function is smooth and flat at zero

Used by

• De rham cohomology is continuous homotopy invariant on smooth manifolds Corollary
• Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
• Weak differentiation has a closed graph on its natural domains Corollary
• A framing, not just the submanifold, determines the Pontryagin-Thom class Counterexample
• A large adverse zero-order term destroys Dirichlet coercivity Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• The H² estimate needs the L² kernel term without injectivity Counterexample
• Collapse of a framed neat cobordism in X times I Definition
• The Pontryagin-Thom map of a framed submanifold Definition
• A nonsymmetric coercive elliptic form Example
• A smooth partition on real space subordinate to two half-spaces Example
• A supported half-twist homeomorphism Example
• Dirichlet Laplacian eigenpairs on an interval Example
• L² forcing defines an H⁻¹ functional Example
• Smoothing a piecewise-linear real-valued function relative to a closed set Example
• The framed unknot represents a generator of pi₃ of S² Example
• The Pontryagin-Thom map of the standard framed equator Example
• The Sobolev weight on a single dyadic annulus Example
• The weak Dirichlet Poisson problem on an interval Example
• A degree-zero sphere map extends over the ball without zeros Lemma
• A manifold bump for a compact set inside an open set Lemma
• A smooth bump between concentric Euclidean balls Lemma
• Compactly supported scaled Euclidean bumps Lemma
• Compatible smooth simplex faces have a neighbourhood extension Lemma
• Deconvolution of a Schwartz function along the dyadic dilates of a fixed kernel Lemma
• Existence of a smooth inhomogeneous dyadic frequency partition Lemma
• Finite ambient partitions near compact sets Lemma
• Finite chart localization gives choice-free integration and compact Stokes Lemma
• Finite chart localization of compactly supported top forms, including boundary Lemma
• Framed points in one component of the frame bundle are framed cobordant Lemma
• Homotopic maps with a common regular value have framed-cobordant preimages Lemma
• Level decomposition of an Hᵖ distribution produces atoms Lemma
• Oppositely framed points cancel in pairs Lemma
• Positively oriented bases of an oriented vector space are path-connected Lemma
• Relative smoothing of a continuous simplex along its faces Lemma
• Schwartz functions with prescribed flatness of the Fourier transform at the origin Lemma
• Smooth representatives of configuration loops Lemma
• Stationary-phase decay for spherical surface measure Lemma
• The collapse of a regular preimage is homotopic to the original map Lemma
• The components of the frame bundle of a connected manifold Lemma

…and 15 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
• Will J. Merry, Differential Geometry (standard reference, not scraped)
• Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)