Manifolds and Tensors

**Need:

1. Want to generalise the flat geometry of special relativity (smooth curved geometries). In flat space, we used inertial coordinates but in a general curved geometry, we need to use arbitrary coordinates.
2. If we think of spacetime as an abstract set of points, a coordinate system can be thought as a map from one part of the spacetime to an open region of the n-dimensional real space. This map is always invertible where it is defined. Hence, when different coordinates systems overlap, they are related by a coordinate transformation, a map ${\tilde{x}}^{\mu }(x^{\nu })$ from an open region in the n-dimensional real space to an another open region in the n-dimensional real space. !! insert picture of maps

• Manifolds
• Vectors
• Cotangent Vectors
• Tensors