Question

Show that a space is locally flat if and only if there exists a local basis of vector ficlds \(\left\\{e_{t} \mid\right.\) that are absolutely parallel, \(D e_{1}=0\).

Short answer

The statement is true. In a locally flat space, around each point, there is a region behaving as a flat space. In this region, we can find a set of absolutely parallel vector fields. Conversely, around a point where a set of absolutely parallel vector fields exists, the space behaves as a flat one, signifying locally flatness of the space at that point.

Step-by-step solution

Understand the definition of 'Locally Flat'

To start with, it is crucial that 'locally flat' is understood. A locally flat space is one where, around any given point, there is a region which behaves just like a flat space. This is a considerable generalization of a flat space.

Understand the definition of 'Absolutely Parallel Vector Fields'

Next, understand what is meant by 'absolutely parallel vector fields'. A vector field is called absolutely parallel if its covariant derivative vanishes. This means, for a set of vector fields {e_{t}}, if D e_{t} = 0 for each t, then this set of vector fields is absolutely parallel.

Proving the if part of the statement

Consider a locally flat space, and pick a point in it. Around this point, there is a region behaving like a flat space. In this region, one can always find a set of vector fields {e_{t}}, which are constant, and hence, their covariant derivative is zero. This means the set of vector fields is absolutely parallel.

Proving the only if part of the statement

Now let's say that there exists a set of absolutely parallel vectors fields {e_{t}} around a point. The behavior of objects around this point will be the same as in a flat space since there's no change in the direction of indices. So this region behaves as if it is a flat space, meaning the space is locally flat at this point.

Key concepts

Locally Flat Spaces

A locally flat space can be imagined as a patch in a more complex surface that resembles a flat plane. Even on a surface as intricate as a wave or a mountain range, there exist small regions around any given point that can be compared to the flatness of a pancake. This concept is particularly useful in differential geometry when examining spaces that are not globally flat.

Locally flat spaces allow for simplifications in mathematical calculations. For instance, complex curvature calculations can be temporarily set aside in such regions, facilitating the analysis of the space's properties. This is pivotal for understanding spaces that don't exhibit flatness all over. It serves as a bridge between fully curved and perfectly flat spaces, enabling mathematicians to apply local Euclidean geometry to non-flat contexts, hence intertwining local and global properties. By understanding and identifying these locally flat regions in a space, mathematicians can apply techniques and results from flat geometry to these small, manageable sections, greatly aiding in the analysis of more complex manifolds.

Vector Fields

Vector fields assign a vector to every point within a given space. Imagine a forest where each tree sways in a specific direction and speed due to the wind. The direction and speed at each tree can be represented as a vector, collectively forming a vector field.

In mathematics, vector fields depict the distribution of vectors across a region of space, such as velocities at each point in a fluid flow or wind speed and direction over a surface. These representations are crucial for analyzing how various physical quantities change across a space.

• Vector fields provide the framework for understanding dynamical systems, fluid dynamics, and electromagnetism.
• They are represented mathematically, often by functions that define each vector's magnitude and direction at every point in the space.
• Understanding vector fields assists in predicting and visualizing field behavior in different scenarios.

When studying differential geometry, vector fields enable the formulation of space's intrinsic properties, assisting in investigations into curvature and geometry.

Covariant Derivative

The covariant derivative is a way to differentiate vector fields in a manner that respects the geometric structure of the space in which the fields exist. Unlike the regular derivative, which only considers changes in value, the covariant derivative takes into account the geometry of the space, adjusting for its curvature and inherent structure.

The covariant derivative is symbolized by the operator \(D\). When we say that a vector field is absolutely parallel, we mean that its covariant derivative is zero (i.e., \(D e_{t} = 0\)). This concept is essential because it indicates that the vector field maintains a constant direction and magnitude throughout the region, reflecting a uniformity akin to traveling along a straight line in flat space.

• Ensures that differentiation aligns with the space's connection.
• Plays a vital role in comparing vectors at different points on a manifold.
• Facilitates calculations involving curvature and motion along surfaces.

Understanding how covariant derivatives operate allows mathematicians to manage the complexities introduced by curvature, making it an indispensable tool for studying and interpreting the behavior of vector fields across varied surfaces.