Riemannian manifolds of sign-definite sectional curvature have been studied by many mathematicians due to the close relationship between the curvature and the topology of a Riemannian manifold.We study Riemannian manifolds whose metric connection is a connection with vectorial torsion. The class of such connections contains the Levi-Civita connection. Although the curvature tensor of such a connection does not possess symmetries of the curvature tensor of the Levi-Civita connection, it is possible to define the sectional curvature. We investigate the question on relations between the sectional curvature of a connection with vectorial torsion and the sectional curvature of the Levi-Civita connection (Riemannian curvature). We also study the sign of sectional curvatures of connections with vectorial torsion. As an example, we consider Lie groups with left-invariant Riemannian metrics.

中文翻译：

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具有矢量扭转的连接的截面曲率

由于曲率与黎曼流形拓扑之间的紧密关系，许多符号学家已经研究了符号定截面曲率的黎曼流形。我们研究了度量连接与矢量扭转有关的黎曼流形。此类连接的类别包含Levi-Civita连接。尽管这种连接的曲率张量不具有Levi-Civita连接的曲率张量的对称性，但是可以定义截面曲率。我们研究了矢量扭转连接的截面曲率与Levi-Civita连接的截面曲率（黎曼曲率）之间的关系的问题。我们还研究了带有矢量扭转的连接截面曲率的符号。举个例子，