  We consider fixed boundary flow with canonical interpretability as principal
components extended on non-linear Riemannian manifolds. We aim to find a flow
with fixed starting and ending points for noisy multivariate data sets lying on
an embedded non-linear Riemannian manifold. In geometric term, the fixed
boundary flow is defined as an optimal curve that moves in the data cloud with
two fixed end points. At any point on the flow, we maximize the inner product
of the vector field, which is calculated locally, and the tangent vector of the
flow. The rigorous definition derives from an optimization problem using the
intrinsic metric on the manifolds. For random data sets, we name the fixed
boundary flow the random fixed boundary flow and analyze its limiting behavior
under noisy observed samples. We construct a high level algorithm to compute
the random fixed boundary flow and the convergence of the algorithm is
provided. We show that the fixed boundary flow yields a concatenate of three
segments, of which one coincides with the usual principal flow when the
manifold is reduced to the Euclidean space. We further prove that the random
fixed boundary flow converges largely to the population fixed boundary flow
with high probability. We illustrate how the random fixed boundary flow can be
used and interpreted, and showcase its application in real data sets.
