Certain Type of Connections & Deformation on Differentiable Manifolds

In 1880 E.B. Christoffel introduced the idea of derivative of one vector field along another vector field under the coordinate transformation which are known by Christffel symbols. In between 1880 Greogorio Ricci-Curbastro and his student Tullio Levi-Civita studied on affine connection. An affine connection is a geometrical object on a differentiable manifold which connects tangent vector field with the nearby tangent vector field. An affine connection can be deffined by many equivalent ways. In an n-dimensional manifold (n > 1); there are infinitely many affine connections. Now we shall consider the definition of affine connection and Levi-Civita connection.