Notes on geometric aspects of effectively hyperbolic critical points when the characteristic roots are real on one side of time

Geometric aspects of effectively hyperbolic critical points on time $t=0$ are discussed assuming that the characteristic roots are real on one side of time t, namely time is positive. In particular, we aim to elucidate the differences in the geometric aspects of effectively hyperbolic critical points on time $t=0$ when the characteristic roots are real on both the positive and negative sides of time.