Differential geometry studies smooth spaces and metrics.
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OpenAI claims that every compact boundaryless smooth Riemannian surface has a smooth length-preserving immersion into R⁴, including nonorientable surfaces.
Read original source Pinned primary manuscript · family 333github.com
Original report: 2026-10-06OpenAI reports the sharp invariant-mass lower bound in terms of minimum enclosing area for one-ended spacetime initial-data exteriors in every dimension n≥3, under the paper’s energy, trapping and decay assumptions.
Read original source Pinned primary manuscript · family 260github.com
Original report: 2026-10-06Differential geometry studies smooth spaces and metrics. An immersion may self-intersect, whereas an embedding cannot; ambient dimension and smoothness are essential.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
OpenAI reports: Smooth isometric immersion of surfaces into four-space
OpenAI reports: The enclosing-area spacetime Penrose inequality
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