Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smoothn-dimensional G-manifold X equipped with a generic G-invariant vector field v,we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\d_k^+X) which takes its values inA(G). Here Ind^G(v) denotes the equivariant index of the field v, {\d_k^+X\}the v-induced Morse stratification (see [M]) of the boundary \d X, and\chi^G(\d_k^+X) the class of the (n - k)-manifold \d_k^+X in $A(G)$. We examinesome applications of this formula to the equivariant real algebraic fields v incompact domains X \subset \R^n defined via a generic polynomial inequality.Next, we link the above formula with the equivariant degrees of certain Gaussmaps. This link is an equivariant generalization of Gottlieb's formulas.
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机译：假设G是一个紧凑的Lie群，A（G）是它的Burnside环。对于配备了通用G不变量向量场v的紧致平滑维G歧管X，我们证明了摩尔斯公式Ind ^ G（v）= \ sum_ {k = 0} ^ {n}（- 1）^ k \ chi ^ G（\ d_k ^ + X），它的值在A（G）中。这里Ind ^ G（v）表示字段v的等变索引，{\ d_k ^ + X \}边界\ d X的v诱导的莫尔斯分层（见[M]），和\ chi ^ G（\ d_k ^ + X）（n-k）流形\ d_k ^ + X在$ A（G）$中的类。我们研究了该公式在通过通用多项式不等式定义的不紧域X \子集\ R ^ n的等变实数代数场中的某些应用。此链接是Gottlieb公式的等变概括。
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