观点

Lemma 4 ([Tangential invariance]). The integrand defining I_J^⊥(V) is the index integrand of V^⊥=V-q_VT . Consequently, for every smooth real function h , I_J^⊥(V+hT)=I_J^⊥(V) provided both fields have the finite integrals required in . …

Lemma 4 ([Tangential invariance]). The integrand defining I_J^⊥(V) is the index integrand of V^⊥=V-q_VT . Consequently, for every smooth real function h , I_J^⊥(V+hT)=I_J^⊥(V) provided both fields have the finite integrals required in . This invariance holds pointwise, without any condition on h at finite endpoints.

证明 1证否 0作为观点被引用 3链条图发起推演

发起推演1

证明2026-10-21

Lemma 3. There are constants δ_0>0 and C<∞ , depending only on the fixed…Proposition 2. The closed smooth three-manifold M defined by admits no smooth…观点

Since D_sT=0 and ⟨V^⊥,T⟩=0 , we have ⟨D_sV^⊥,T⟩=0 and |D_sV|^2=|D_sV^⊥|^2+| q_V|^2. The curvature symmetries give ⟨R(V,T)T,V⟩= ⟨R(V^⊥,T)T,V^⊥⟩ . Both assertions follow.

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