Abstract
I prove the bistability of linear evolution equations \(x'=A(t)x\) in a Banach space E, where the operator-valued function A is of the form \(A(t)=f'(t)G(t,f(t))\) for a binary operator-valued function G and a scalar function f. The constant that bounds the solutions of the equation is computed explicitly; it is independent of f, in a sense. Two geometric applications of the stability result are presented. Firstly, I show that the parallel transport along a curve \(\gamma \) in a manifold, with respect to some linear connection, is bounded in terms of the length of the projection of \(\gamma \) to a manifold of one dimension lower. Secondly, I prove an extendability result for parallel sections in vector bundles, thus answering a question by Antonio J. Di Scala.
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Notes
I suggest that some research in that direction be carried out.
I omit the details yielding this as well as the preceding assertion.
I omit the details.
Note that the value of the function A at \(a_n\) is arbitrary.
a is what is called a partition of I.
Note that L is the same thing as the total variation \(V_F\), just with a different name.
The simple connectedness can be somewhat weakened.
Note that in ZF set theory, f, as a set, is the graph of f.
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Acknowledgments
I simply must thank Antonio J. Di Scala, for he keeps bringing beautiful mathematics to people (like me).
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Kirschner, T. Uniform Stability of Linear Evolution Equations with Applications to Parallel Transports. Differ Equ Dyn Syst 27, 561–584 (2019). https://doi.org/10.1007/s12591-016-0294-x
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DOI: https://doi.org/10.1007/s12591-016-0294-x
Keywords
- Parallel transport
- Linear evolution equation
- Linear differential equation
- Bistability
- Uniform stability
- Bounded solutions