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Modern differential geometry is based on two fundamental theories. The first emerged around 1850 with the revolutionary ideas of Bernhard Riemann and later evolved into what is now called Riemannian geometry. The second emerged around 1870 with the revolutionary ideas of Sophus Lie and is now called Lie theory. In this talk, we will explain that the concept of covariant di§erentiation, in particular the concepts of torsion and curvature, which originated in Riemannian geometry, emerge also from Lie theory, but with quite di§erent geometric interpretations of torsion and curvature. If time permits, we will indicate some surprising consequences of this new derivation. This talk is a summary of Part I of the book ''An Alternative Approach to Lie Groups and Geometric Structures'' to be published by OUP in July.