The Hessian Metric of Two Classes of Homogeneous Polynomials on R<sup>3</sup>

本文主要对Hessian度量诱导的截面曲率展开分析。首先阐述Hessian度量和与之相关的Christoffor符号和曲率张量公式。接着介绍当定义域U为锥时，齐次函数的相关概念，以及Clebsch covariant S(f)和截面曲率的联系。利用R3中开子集U和超平面M = {f = 1}相切2-平面上一点处的截面曲率可由S(f)和 Hessian 行列式H(f)表示，其中f为R3上的齐次多项式，得到不变量S(f)为零与Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)方程等价的条件，即f可以表示为两种特殊的形式。This paper primarily analyzes the sectional curvature induced by the Hessian metric. It begins by detailing the Hessian metric and its associated Christoffel symbols and curvature tensor formulas. It then introduces the concept of homogeneous functions when the domain U is a cone, as well as the relationship between the Clebsch covariant S(f) and the sectional curvature. Using an open subset U in R3 and a hypersurface M = {f = 1}, the sectional curvature at a point on a 2-plane tangent to M can be expressed in terms of S(f) and the Hessian matrix H(f), where f is a homogeneous polynomial on R3. The condition for the invariant S(f) to be zero is equivalent to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, which implies that f can be represented in two specific forms.