**Core Concept:** Spinal cord infarction is a rare condition that occurs when there is an acute occlusion of an artery supplying blood to the spinal cord, leading to infarction and subsequent neurological deficits. In this scenario, the patient has spinal infarction affecting the posterior white columns in the tenth thoracic segment of the spinal cord, resulting in loss of certain sensory functions but sparing motor functions.

**Why the Correct Answer is Right:** In spinal cord infarction, the posterior columns are predominantly affected due to their rich supply from posterior spinal arteries. These posterior columns are responsible for the sensory functions related to vibration, position, and joint movement sense. In this case, the sensory loss is below the umbilicus, indicating the infarction in the thoracic spinal cord, where the posterior columns are involved. Motor functions are spared because the anterior spinal artery supplies the anterior horns responsible for motor functions.

**Why Each Wrong Option is Incorrect:**

A. Loss of vibration sense: Since the posterior columns are involved in the infarction, loss of vibration sense would be expected, which is not mentioned in the options.

B. Loss of touch and pain sensation: These sensory functions are also primarily affected in posterior column involvement, but the correct answer focuses on specific functions that are preserved.

C. Preserved motor functions: Motor functions are spared due to the anterior spinal artery supply to the anterior horns, not mentioned in the options.

D. Preserved deep tendon reflexes: Deep tendon reflexes are a motor function and would not be spared in this scenario.

**Clinical Pearl:** This case highlights the importance of understanding the spinal cord anatomy and the specific functions of the anterior and posterior columns. Recognizing the preserved sensory functions can help differentiate between spinal cord infarction and other causes of sensory loss like radiculopathy or peripheral neuropathy.

**Correct Answer:** Loss of vibration sense (C) is the correct answer as it represents a preserved function in the context of posterior column involvement.