# A finite-dimensionality result

Chapter

## Abstract

As briefly mentioned at the beginning of the previous chapter, typical geometric applications of Theorem 4.5 are obtained by applying it when the function where

*ψ*is the norm of the section of a suitable vector bundle. In appropriate circumstances, the theorem guarantees that certain vector subspaces of such sections are trivial, the main geometric assumption being the existence of a positive solution*ϕ*of the differential inequality$$
\Delta \varphi + Ha\left( x \right)\varphi \leqslant 0 weakly on M,
$$

(5.1)

*a*(*x*) is a lower bound for the relevant curvature term. According to Lemma 3.10 this amounts to requiring that the bottom of the spectrum of the Schrödinger operator −Δ −*Ha*(*x*) is non-negative.## Keywords

Riemannian Manifold Vector Bundle Harnack Inequality Morse Index Complete Riemannian Manifold
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