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Volume of a small Extrinsic Ball in a Submanifold
Author(s) -
Karp Leon,
Pinsky Mark
Publication year - 1989
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/21.1.87
Subject(s) - submanifold , mathematics , ball (mathematics) , quadratic equation , mathematical analysis , invariant (physics) , term (time) , spheres , curvature , pure mathematics , geometry , mathematical physics , physics , quantum mechanics , astronomy
For a submanifold M p ↪ R , we determine a two‐term asymptotic formula for vol ( M p ∩ B ∈ ( x )) for x ∈ M p as ∈ ↓ 0. The second term is a quadratic curvature invariant of the second fundamental form of the imbedding. Imbedded spheres are characterized among compact hypersurfaces by this term.