| He BAIHE.[J].数学年刊A辑,1981,2(2):233~242 |
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| CONTINUOUS MAPPING FROM SPHERESTO EUCLIDEAN SPACES |
| Received:December 14, 1979 |
| DOI： |
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| In the forties Knaster, B., posed the following problem: Gieven a continuous
mapping f of an (m+n-2) sphere \({S^{m + n - 2}}\) into the Euclidean m -space \({R^m}\） and n distinct points it \({u_1}, \cdots {u_n}\） of \({S^{m + n - 2}}\); does there exist a rotation r such that \[f(r{u_1}) = \cdots = f(r{u_n})?\] In this paper, the index under periodic transfromation of StieM manifold is applied to prove the following theorem:
Given a continuous mapping \（f:{S^{k - 1}} \to {R^m}\）， n distinct points \({u_1}, \cdots {u_n} \in {S^{k - 1}}\）
viewed as unit vectors satisfying \({u_i}{u_j} = {u_{i + 1}}{u_{j + 1}},i,j \in {I_n}\), and suppose\({u_1}, \cdots {u_n}\） have rank l, then in each of the following cases, there is a!rotation r such that \[f(r{u_1}) = \cdots = f(r{u_n})\]
1. \[n \ne 2,3,k - 1 = (n - 1)m\];
2. n is an odd prime number, l even,\[k - 1 = \left[ {\frac{{(n - 1)m}}{2}} \right] + l - 2\];
3. n is an odd prime number, l odd, \[l \ge \left[ {\frac{{(n - 1)m}}{2}} \right] + 1,k - 1 = \left[ {\frac{{(n - 1)m}}{2}} \right] + l - 2;\]
4. n is an odd prime number, l odd, \[l < \left[ {\frac{{(n - 1)m}}{2}} \right] + 1,k - 1 = (n - 1)m + 1;\] where [*] is the least even number>*.
This theorem generalizes the classical Borsuk-Ulam theorem. |
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