Differential geometry on simplicial spaces

Penna, Michael A.

doi:10.1090/s0002-9947-1975-0391146-5

Cited by 6 publications

(12 citation statements)

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“…()• Introduction* The basic theme of (6) and (7) is that there is a striking similarity between the geometry of smoth manifolds and the geometry of simplicial complexes. The purpose of this paper is to continue this theme for smooth manifolds and combinatorial manifolds.…”

Section: On the Geometry Of Combinatorial Manifolds Michael A Pennamentioning

confidence: 99%

“…See (1) and (9) for related definitions.) Section 1 is devoted to a brief review of some of the terminology and results of (6) and (7). The goal of §2 is the characterization of continuous vector fields on combinatorial manifolds.…”

Section: On the Geometry Of Combinatorial Manifolds Michael A Pennamentioning

confidence: 99%

“…Before proving Proposition 2.3, let us first recall (see [6]) that one can think of elements / of the ring A(U) of piecewise smooth real-valued functions on U as compatible tuples (f a ) e X α A(i7 α ) of smooth real-valued functions f a e A( U a ) defined on the wedges U a Q U; here "compatible" means that if U a and U β are wedges of U then fa\u a nu β = fβ\u a nu β If functions are expressed in this manner then for > -(/ β ) and 0 -(</ α ) in A(C7), f + g = (f a + g a ) and /. ff = (f a .g a ).…”

Section: <And^(m) A(m)) Of A(m)-linear Alternating Maps From X Q <S?(m) To A(m) and (4) If D: λ*E{m) -* λ*E(m) Is The Differential Of Thementioning

confidence: 99%

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On the geometry of combinatorial manifolds

Penna¹

1978

Pacific J. Math.

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Section: On the Geometry Of Combinatorial Manifolds Michael A Pennamentioning

confidence: 99%

Section: On the Geometry Of Combinatorial Manifolds Michael A Pennamentioning

confidence: 99%

Section: <And^(m) A(m)) Of A(m)-linear Alternating Maps From X Q <S?(m) To A(m) and (4) If D: λ*E{m) -* λ*E(m) Is The Differential Of Thementioning

confidence: 99%

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On the geometry of combinatorial manifolds

Penna¹

1978

Pacific J. Math.

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“…The theory of simplicial bundles is the dual of a corresponding bundle theory which is used in studying piecewise smooth forms on polyhedra (see [11]); some of the terminology and results of [11] are briefly reviewed in §1. Simplicial bundles are defined in §2: every polyhedron has a tangent object in the category of simplicial bundles in much the same way every smooth manifold has a tangent object in the category of smooth vector bundles.…”

mentioning

confidence: 99%

“…Simplicial spaces. This section is devoted to a brief review of some of the terminology and results of [11].…”

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confidence: 99%

Vector fields on polyhedra

Penna¹

1977

Trans. Amer. Math. Soc.

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Abstract. This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra^ and applications.Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms. 0. Introduction. The purpose of this paper is to describe a bundle theory for studying vector fields and their integral flows on polyhedra and, in particular, on combinatorial manifolds. Two applications of this theory are presented; these results are motivated by corresponding results for smooth manifolds.Here are the results obtained:Theorem A. The sum of the indices of any integrable vector field on a polyhedron P which has a finite number of zeroes is the Euler characteristic x(P) of P. Furthermore, if M is a closed combinatorial manifold with x(M) = 0, then there is a nowhere vanishing integrable vector field on M.Theorem A corresponds to a classical result for smooth manifolds due to Hopf (see [10]). Actually Hopfs original result (see [7]) is also valid for a particular type of vector field on a combinatorial manifold. It will be seen that even though there are similarities between Hopfs original result and Theorem A, there are also major differences. As a special case of Theorem A one also obtains an analog to a calculation of Whitney (see [14]) giving the Euler characteristic of a triangulated smooth manifold using the singularities of vector fields. Now let M be a closed, oriented, 2/i-dimensional combinatorial manifold.

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